This page lists a core term of calculus. The term is used widely, and a thorough understanding of its definition is critical. See a complete list of core terminology
The term "limit" in mathematics is closely related to one of the many senses in which the term "limit" is used in day-to-day English. In day-to-day English, there are two uses of the term "limit":
Limit as something that one approaches, or is headed toward
Limit as a boundary or cap that cannot be crossed or exceeded
The mathematical term "limit" refers to the first of these two meanings. In other words, the mathematical concept of limit is a formalization of the intuitive concept of limit as something that one approaches or is headed toward.
For a function , the notation:
is meant to say "the limit, as approaches , of the function value " and thus, the mathematical equality:
is meant to say "the limit, as approaches , of the function value , is ." In a rough sense, what this means is that as gets closer and closer to , eventually comes, and stays, close enough to .
Graphical interpretation
The graphical interpretation of "" is that, if we move along the graph of the function in the plane, then the graph approaches the point whether we make approach from the left or the right. However, this interpretation works well only if is continuous on the immediate left and immediate right of .
This interpretation is sometimes termed the "two finger test" where one finger is used to follow the graph for slightly less than and the other finger is used to follow the graph for slightly greater than .
The interpretation is problematic in that it is not really a definition, and fails to have computational utility for wildly oscillatory functions or functions with other forms of weird behavior.
Two key ideas
The concept of limit involves two key ideas, both of which help explain why the definition is structured the way it is:
Arbitrarily close: The limit depends on how things behave arbitrarily close to the point involved. The notion of "arbitrarily close" is difficult to quantify non-mathematically, but what it means is that any fixed distance is too much. For instance, if doing , we can take points close to 2 such as 2.1, 2.01, 2.001, 2.0001, 2.0000001, 2.000000000000001. Any of these points, viewed in and of itself, is too far from 2 to offer any meaningful information. It is only the behavior in the limit, as we get arbitrarily close, that matters.
Trapping of the function close by: For a function to have a certain limit at a point, it is not sufficient to have the function value come close to that point. Rather, for to hold, it is necessary that for very close to , the function value is trapped close to . It is not enough that it keeps oscillating between being close to and being far from .
Vipul: Okay, so in this talk, I'm going to
go over the basic
0:00:19.259,0:00:24.619
motivation behind the definition of limit,
and not so much the
0:00:24.619,0:00:28.099
epsilon-delta definition. This is just an intuitive idea,
and a few somewhat
0:00:28.099,0:00:29.680
non-intuitive aspects of that.
0:00:29.680,0:00:36.680
Here I have the notation: "limit as x approaches
c of f(x) is L" is
0:00:37.540,0:00:42.079
written like this. Limit ... Under the limit,
we write where the
0:00:42.079,0:00:46.180
domain point goes, so x is approaching a value,
c, and c could be an
0:00:46.180,0:00:51.059
actual number. x, however, will always be
a variable letter. This x
0:00:51.059,0:00:54.519
will not be a number. c could be a number
like zero, one, two, three,
0:00:54.519,0:00:55.329
or something.
0:00:55.329,0:01:02.050
f(x). f is the function. We are saying that
as x approaches some
0:01:02.050,0:01:06.640
number c, f(x) approaches some number L, and
that's what this is:
0:01:06.640,0:01:09.030
Limit as x approaches c of f(x) is L.
0:01:09.030,0:01:15.259
Now what does this mean? Roughly what it means
is that as x is coming
0:01:15.259,0:01:22.259
closer and closer to c, f(x) is sort of hanging
around L. It's coming
0:01:22.410,0:01:28.720
closer and closer to L. By the way, there
are two senses in which the
0:01:28.720,0:01:32.429
word limit is used in the English language:
One meaning is limit in
0:01:32.429,0:01:36.310
this approach sense, which is the mathematical
meaning of limit.
0:01:36.310,0:01:41.319
There is another sense in which the word limit
is used in the English
0:01:41.319,0:01:46.220
language, which is limit as a boundary or as a cap or as a bound.
0:01:46.220,0:01:53.160
We may say, there is a limit to how many apples
you can eat from the
0:01:53.160,0:01:58.640
fruit bowl or something, and that sense of
limit is not used ... for
0:01:58.640,0:02:02.110
that sense of limit you do not use the word
"limit" in mathematics. For
0:02:02.110,0:02:05.899
that sense of limit, you use the word bound.
In mathematics, we
0:02:05.899,0:02:11.800
reserve the use of the word limit only for
this approach sense. Just
0:02:11.800,0:02:18.800
so we don't get confused in mathematics.
As I said, the idea is that
0:02:21.120,0:02:25.760
as x approaches c, f(x) approaches L, so as
x is coming closer and
0:02:25.760,0:02:29.480
closer to c, the distance between x and c
is becoming smaller and
0:02:29.480,0:02:32.740
smaller, the distance between f(x) and L is
also roughly becoming
0:02:32.740,0:02:37.980
smaller and smaller. This doesn't quite
work unless your function is
0:02:37.980,0:02:41.250
increasing or decreasing near c, so you could
have various
0:02:41.250,0:02:46.750
complications with oscillatory functions,
so the point is this notion
0:02:46.750,0:02:52.170
doesn't really ... it's not very clear what
we mean here without further
0:02:52.170,0:02:55.470
elaboration and without a clear definition.
0:02:55.470,0:03:02.470
I'm going to sort of move up toward the definition,
and before we go
0:03:02.970,0:03:09.180
there, I want to say, that there is a graphical
concept of limit,
0:03:09.180,0:03:13.430
which you may have seen in school. (well,
if you've seen limits in
0:03:13.430,0:03:17.110
school, which hopefully you have. This video
is sort of more of a
0:03:17.110,0:03:21.500
review type than learning it for the first
time). Let's try to
0:03:21.500,0:03:24.630
understand this from that point of view.
0:03:24.630,0:03:31.630
Let's say, you have a function whose graph
looks something like this.
0:03:35.990,0:03:42.990
This is x is c, so this is the value x is
c, and this is the graph of
0:03:44.069,0:03:48.310
the function, these curves are the graph of
the function, so where x
0:03:48.310,0:03:53.900
is less than c, the graph is along this curve.
For x greater than c,
0:03:53.900,0:03:58.120
the graph is this curve. So x less than c,
the graph is this curve; x
0:03:58.120,0:04:01.740
greater than c, the graph is this curve. At
x equal to c, the value
0:04:01.740,0:04:06.330
is that filled dot.
0:04:06.330,0:04:13.330
You can see from here that as x is approaching
c from the left, so if
0:04:13.880,0:04:17.819
you take values of x, which are slightly less
than c, the function
0:04:17.819,0:04:23.259
values ... so the function, the graph of it,
the function values are
0:04:23.259,0:04:27.449
their respective y coordinates, so this is
x, this is y, this is the
0:04:27.449,0:04:34.449
graph. y is f(x). When x is to the immediate
left of c, the value, y
0:04:35.749,0:04:42.749
value, the y equals f(x) value is ... are
these values, so this or
0:04:44.610,0:04:51.610
this. As x approaches c from the left, the
y values are approaching
0:04:53.699,0:04:57.240
the y coordinate of this open circle.
0:04:57.240,0:05:04.240
In a sense, if you just were looking at the
limit from the left for x
0:05:05.680,0:05:10.830
approaching c from the left, then the limit
would be the y coordinate
0:05:10.830,0:05:16.279
of this open circle. You can also see an x
approaches c from the
0:05:16.279,0:05:22.749
right, so approaches from here ... the y coordinate
is approaching the y
0:05:22.749,0:05:29.749
coordinate of this thing, this open circle
on top. There are actually
0:05:31.009,0:05:38.009
two concepts here, the left-hand limit
is this value. We will call
this L1. The right-hand limit is this value,
0:05:45.599,0:05:49.349
L2, so the left-hand
limit, which is the notation as limit as x
0:05:49.349,0:05:56.349
approaches c from the left
of f(x) is L1, the right-hand limit from the
0:05:58.089,0:06:05.089
right, that's plus of f(x),
is L2, and the value f of c is some third
0:06:08.059,0:06:15.059
number. We don't know what
it is, but f of c, L1, L2, are in this case
0:06:16.770,0:06:18.360
all different.
0:06:18.360,0:06:25.360
What does this mean as far as the limit is
concerned? Well, the
0:06:25.900,0:06:28.259
concept of limit is usually a concept of two
sided limit, which
0:06:28.259,0:06:33.419
means that in this case the limit as x approaches
c of f(x) does not
0:06:33.419,0:06:36.289
exist because you have a left-hand limit,
and you have a right-hand
0:06:36.289,0:06:39.860
limit, and they are not equal to each other.
The value, as such,
0:06:39.860,0:06:43.279
doesn't matter, so whether the value exists,
what it is, does not
0:06:43.279,0:06:46.379
affect this concept of limit, but the real
problem here is that the
0:06:46.379,0:06:48.490
left-hand limit and right-hand limit are not
equal. The left-hand
0:06:48.490,0:06:55.490
limit is here; the right-hand limit is up
here.
0:06:59.050,0:07:03.499
This graphical interpretation, you see the
graphical interpretation is
0:07:03.499,0:07:07.749
sort of that: for the left-hand limit, you
basically sort of follow
0:07:07.749,0:07:11.499
the graph on the immediate left and see where
it's headed to and you
0:07:11.499,0:07:15.789
get the y coordinate of that. For the right-hand
limit, you follow
0:07:15.789,0:07:21.129
the graph on the right and see where we're
headed to, and get the y
0:07:21.129,0:07:22.240
coordinate of that.
0:07:22.240,0:07:29.240
Let me make an example, where the limit does
exist. Let's say you
0:07:42.899,0:07:48.449
have a picture, something like this. In this
case, the left-hand limit
0:07:48.449,0:07:52.610
and right-hand limit are the same thing, so
this number, but the
0:07:52.610,0:07:55.889
value is different. You could also have
a situation where the value
0:07:55.889,0:08:00.460
doesn't exist at all. The function isn't
defined at the point, but
0:08:00.460,0:08:03.139
the limit still exists because the left-hand
limit and right-hand
0:08:03.139,0:08:04.719
limit are the same.
0:08:04.719,0:08:09.979
Now, all these examples, there's sort of a
crude way of putting this
0:08:09.979,0:08:13.710
idea, which is called the two-finger test.
You may have heard it in
0:08:13.710,0:08:18.399
some slightly different names. The two-finger
test idea is that you
0:08:18.399,0:08:23.929
use one finger to trace the curve on the immediate
left and see where
0:08:23.929,0:08:28.259
that's headed to, and use another finger
to trace the curve on the
0:08:28.259,0:08:33.640
immediate right and see where that's headed
to, and if your two
0:08:33.640,0:08:38.270
fingers can meet each other, then the place
where they meet, the y
0:08:38.270,0:08:41.870
coordinate of that, is the limit. If, however,
they do not come to
0:08:41.870,0:08:46.940
meet each other, which happens in this case,
one of them is here, one
0:08:46.940,0:08:51.120
is here, and then the limit doesn't exist
because the left-hand limit
0:08:51.120,0:08:53.509
and right-hand limit are not equal.
0:08:53.509,0:08:59.819
This, hopefully, you have seen in great detail
when you've done
0:08:59.819,0:09:05.779
limits in detail in school. However, what
I want to say here is that
0:09:05.779,0:09:11.850
this two-finger test is not really a good
definition of limit. What's
0:09:11.850,0:09:13.600
the problem? The problem is that you could
have really crazy
0:09:13.600,0:09:18.790
function, and it's really hard to move your
finger along the graph of
0:09:18.790,0:09:25.220
the function. If the function sort of jumps
around a lot, it's really
0:09:25.220,0:09:29.440
hard, and it doesn't really solve any problem.
It's not really a
0:09:29.440,0:09:35.100
mathematically pure thing. It's like trying
to answer the
0:09:35.100,0:09:39.540
mathematical question using a physical description,
which is sort of
0:09:39.540,0:09:41.579
the wrong type of answer.
0:09:41.579,0:09:45.610
While this is very good for a basic intuition
for very simple types of
0:09:45.610,0:09:50.040
functions, it's not actually the correct idea
of limit. What kind of
0:09:50.040,0:09:56.990
things could give us trouble? Why do we need
to refine our
0:09:56.990,0:10:03.209
understanding of limit? The main thing is
functions which have a lot
0:10:03.209,0:10:07.980
of oscillation. Let me do an example.
0:10:07.980,0:10:14.980
I'm now going to write down a type of function
where, in fact, you
0:10:18.220,0:10:21.899
have to develop a clear cut concept of limit
to be able to answer this
0:10:21.899,0:10:28.899
question precisely. This is a graph of a function,
sine 1 over x.
0:10:28.959,0:10:32.920
Now this looks a little weird. It's not 1
over sine x; that would
0:10:32.920,0:10:39.920
just equal cosecant x. It's not that. It's sine
of 1 over x, and this
0:10:44.879,0:10:50.220
function itself is not defined at x equals
zero, but just the fact
0:10:50.220,0:10:52.660
that that's not defined, isn't good enough
for us to say the limit
0:10:52.660,0:10:55.139
doesn't exist; we actually have
to try to make a picture
0:10:55.139,0:10:57.660
of this and try to understand what the limit
is going to be.
0:10:57.660,0:11:04.660
Let's first make the picture of sine x. Sine-x
looks like that. How
0:11:12.560,0:11:19.560
will sine 1 over x look? Let's start off where
x is nearly infinity.
0:11:20.100,0:11:25.759
When x is very large positive, 1 over x is
near zero, slightly
0:11:25.759,0:11:30.660
positive, just slightly bigger than zero,
and sine 1 over x is
0:11:30.660,0:11:36.879
therefore slightly positive. It's like here.
It's going to start off
0:11:36.879,0:11:42.810
with an asymptote, a horizontal asymptote, at zero.
Then it's going to sort of go
0:11:42.810,0:11:49.420
this path, but much more slowly, each one,
then it's going to go this
0:11:49.420,0:11:56.420
path, but in reverse, so like that. Then it's
going to go this path,
0:11:57.149,0:12:00.740
but now it does all these oscillations, all
of these oscillations. It
0:12:00.740,0:12:03.569
has to go faster and faster.
0:12:03.569,0:12:10.569
For instance, this is pi, this 1 over pi,
then this is 2 pi, this
0:12:12.329,0:12:16.990
number is 1 over 2 pi, then the then next
time it reaches zero will be
0:12:16.990,0:12:21.160
1 over 3 pi, and so on. What's going to
happen is that near zero it's
0:12:21.160,0:12:24.579
going to be crazily oscillating between minus
1, and 1. The frequency
0:12:24.579,0:12:29.170
of the oscillation keeps getting faster and
faster as you come closer
0:12:29.170,0:12:34.050
and closer to zero. The same type of picture
on the left side as
0:12:34.050,0:12:40.360
well; it's just that it's an odd function.
It's this kind of picture.
0:12:40.360,0:12:47.360
I'll make a bigger picture here ... I'll make
a bigger picture on another
0:12:53.649,0:13:00.649
one. all of these oscillation should be between
minus 1 and 1, and we
0:13:22.439,0:13:29.399
get faster so we get faster and faster, and
now my pen is too thick.
0:13:29.399,0:13:31.600
It's the same, even if you used your finger
instead of the pen to
0:13:31.600,0:13:38.600
place it, it would be too thick, it's called
the thick finger problem.
0:13:38.850,0:13:45.060
I'm not being very accurate here, but just
the idea. The pen or
0:13:45.060,0:13:49.199
finger is too thick, but actually, there's
a very thin line, and it's
0:13:49.199,0:13:52.519
an infinitely thin line of the graph, which
goes like that.
0:13:52.519,0:13:59.519
Let's get back to our question: What is limit
as x approaches zero,
0:14:02.699,0:14:09.699
sine 1 over x. I want you to think about this
a bit. Think about like
0:14:13.439,0:14:18.050
the finger test. You move your finger around,
move it like this,
0:14:18.050,0:14:21.579
this, this ... you're sort of getting close
to zero but still not quite
0:14:21.579,0:14:28.579
reaching it. It's ... where are you headed?
It's kind of a little
0:14:31.610,0:14:36.879
unclear. Notice, it's not that just because
we plug in zero doesn't
0:14:36.879,0:14:39.170
make sense, the limit doesn't... That's
not the issue. The issue is
0:14:39.170,0:14:43.249
that after you make the graph, it's unclear
what's happening.
0:14:43.249,0:14:49.329
One kind of logic is that, yeah, the limit
is zero? Why? Well, it's
0:14:49.329,0:14:52.949
kind of balanced around zero, right? It's a bit
above and below, and it keeps
0:14:52.949,0:14:59.949
coming close to zero. That any number of the
form x is 1 over N pi,
0:15:00.329,0:15:07.329
sine 1 over x is zero. It keeps coming close
to zero. As x
0:15:07.990,0:15:12.459
approaches zero, this number keeps coming
close to zero.
0:15:12.459,0:15:17.449
If you think of limit as something it's
approaching, then as x
0:15:17.449,0:15:24.449
approaches zero, sine 1 over x is sort of
coming close to zero, is it?
0:15:31.230,0:15:36.550
It's definitely coming near zero, right? Anything
you make around
0:15:36.550,0:15:41.920
zero, any small ... this you make around zero,
the graph is going to
0:15:41.920,0:15:42.399
enter that.
0:15:42.399,0:15:47.269
On the other hand, it's not really staying
close to zero. It's kind of
0:15:47.269,0:15:50.300
oscillating within [-1,1]. However
small an interval you
0:15:50.300,0:15:54.540
take around zero on the x thing, the function
is oscillating between
0:15:54.540,0:15:57.600
minus 1 and 1. It's not staying faithful to
zero.
0:15:57.600,0:16:02.249
Now you have kind of this question: What should
be the correct
0:16:02.249,0:16:09.249
definition of this limit? Should it mean that
it approaches the
0:16:10.029,0:16:15.100
point, but maybe goes in and out, close and
far? Or should it mean it
0:16:15.100,0:16:18.879
approaches and stays close to the point? That
is like a judgment you
0:16:18.879,0:16:22.629
have to make in the definition, and it so
happens that people who
0:16:22.629,0:16:28.639
tried defining this chose the latter idea;
that is, it should come
0:16:28.639,0:16:33.089
close and stay close. So that's actually
key idea number two we have
0:16:33.089,0:16:38.290
here the function ... for the function to
have a limit at the point, the
0:16:38.290,0:16:43.639
function needs to be trapped near the limit,
close to the point in the
0:16:43.639,0:16:45.079
domain.
0:16:45.079,0:16:49.459
This is, therefore, it doesn't have a limit
at zero because the
0:16:49.459,0:16:54.420
function is oscillating too widely. You cannot
trap it. You cannot
0:16:54.420,0:17:01.059
trap the function values. You cannot say that...
you cannot trap the
0:17:01.059,0:17:08.059
function value, say, in this small horizontal
strip near zero. You
0:17:08.319,0:17:11.650
cannot trap in the area, so that means the
limit cannot be zero, but
0:17:11.650,0:17:15.400
the same logic works anywhere else. The limit
cannot be half, because
0:17:15.400,0:17:20.440
you cannot trap the function in a small horizontal
strip about half
0:17:20.440,0:17:22.130
whereas x approaches zero.
0:17:22.130,0:17:26.440
We will actually talk about this example in
great detail in our future
0:17:26.440,0:17:30.330
with you after we've seen the formal definition,
but the key idea you
0:17:30.330,0:17:33.890
need to remember is that the function doesn't
just need to come close
0:17:33.890,0:17:37.340
to the point of its limit. It actually needs
to stay close. It needs
0:17:37.340,0:17:41.050
to be trapped near the point.
0:17:41.050,0:17:44.810
The other important idea regarding limits
is that the limit depends
0:17:44.810,0:17:50.370
only on the behavior very, very close to the
point. What do I mean by
0:17:50.370,0:17:56.580
very, very close? If you were working it like,
the real goal, you may
0:17:56.580,0:18:02.300
say, it's like, think of some really small
number and you say that
0:18:02.300,0:18:07.050
much distance from it. Let's say I want to
get the limit as x
0:18:07.050,0:18:14.050
approaches 2...I'll just write it here. I
want to get, let's say,
0:18:23.520,0:18:30.520
limit has x approaches 2 of some function,
we may say, well, we sort
0:18:30.550,0:18:37.550
of ... what's close enough? Is 2.1 close
enough? No, that's too far.
0:18:38.750,0:18:43.380
What about 2.0000001? Is that close enough?
0:18:43.380,0:18:47.420
Now, if you weren't a mathematician, you
would probably say, "Yes,
0:18:47.420,0:18:54.420
this is close enough." The difference is like
... so it's
0:18:57.040,0:19:04.040
10^{-7}. It's really only close to 2 compared
to our usual sense of
0:19:12.990,0:19:16.670
numbers, but as far as mathematics is concerned,
both of these numbers
0:19:16.670,0:19:21.110
are really far from 2. Any individual number
that is not 2 is very
0:19:21.110,0:19:22.130
far from 2.
0:19:22.130,0:19:29.130
What do I mean by that, well, think back to
one of our
0:19:29.670,0:19:36.670
pictures. Here's a picture. Supposed I take
some points. Let's say
0:19:41.970,0:19:47.640
this is 2, and suppose I take one point here,
which is really close to
0:19:47.640,0:19:50.970
2, and I just change the value of the function
at that point. I
0:19:50.970,0:19:55.200
change the value of the function at that point,
or I just change the
0:19:55.200,0:19:59.990
entire picture of the graph from that point
rightward. I just take
0:19:59.990,0:20:05.940
this picture, and I change it to, let's say
... so I replace this
0:20:05.940,0:20:11.410
picture by that picture, or I replace the
picture by some totally new
0:20:11.410,0:20:15.250
picture like that picture. I just change the
part of the graph to the
0:20:15.250,0:20:21.440
right of some point, like 2.00001, whatever.
Will that effect the
0:20:21.440,0:20:25.770
limit at 2? No, because the limit at 2 really
depends only on the
0:20:25.770,0:20:27.520
behavior if you're really, really close.
0:20:27.520,0:20:32.040
If you take any fixed point, which is not
2, and you change the
0:20:32.040,0:20:35.000
behavior, sort of at that point or
farther away than that
0:20:35.000,0:20:42.000
point, then the behavior close to 2 doesn't
get affected. That's the
0:20:42.820,0:20:46.660
other key idea here. Actually I did these
in reverse order.
0:20:46.660,0:20:52.060
That's how it was coming naturally, but I'll
just say it again.
0:20:52.060,0:20:56.570
The limit depends on the behavior arbitrarily
close to the point. It
0:20:56.570,0:21:00.210
doesn't depend on the behavior at any single
specific other point. It
0:21:00.210,0:21:06.910
just depends on the behavior as you approach
the point and any other
0:21:06.910,0:21:11.330
point is far away. It's only sort of together
that all the other
0:21:11.330,0:21:16.230
points matter, and it's only them getting
really close that
0:21:16.230,0:21:19.790
matters. The other thing is that the function
actually needs to be
0:21:19.790,0:21:26.790
trapped near the point for the limit notion
to be true. This type of
0:21:26.860,0:21:29.650
picture where it's oscillating between minus
1 and 1, however close
0:21:29.650,0:21:35.150
you get to zero, keeps oscillating, and so
you cannot trap it around
0:21:35.150,0:21:40.590
any point. You cannot trap the function value
in any small enough
0:21:40.590,0:21:47.590
strip. In that case, the limit doesn't exist.
In subsequent videos,
0:21:48.550,0:21:54.630
we'll see the epsilon delta definition, we'll do a bit
of formalism to that, and
0:21:54.630,0:22:00.640
then we'll come back to some of these issues
later with the formal
0:22:00.640,0:22:01.870
understanding.
Checkpoint questions:
To figure out the limit of a function at , does the value of the function at matter? Does the value of the function at matter? ? How close is close enough?
What is the limit ? What's the intuitive idea behind the reasoning? More formal versions of this reasoning will be introduced after we have seen the definition.
Definition for finite limit for function of one variable
Two-sided limit
Suppose is a function of one variable and is a point such that is defined to the immediate left and immediate right of (note that may or may not be defined at ). In other words, there exists some value such that is defined on .
For a given value , we say that:
if the following holds:
For every , there exists such that for all satisfying , we have .
The definition is broken down into its four clauses below:
Clause
Interval description
Symbol explanations
For every
The symbol is a Greek lowercase letter pronounced "epsilon". Although the definition customarily uses the letter , it can be replaced by any other letter, as long as the letter is different from the other letters in use. The reason for sticking to a standard letter choice is that it reduces cognitive overload.
there exists such that
The symbol is a Greek lowercase letter pronounced "delta" Although the definition customarily uses the letter , it can be replaced by any other letter, as long as the letter is different from the other letters in use. The reason for sticking to a standard letter choice is that it reduces cognitive overload.
for all satisfying
The symbol stands for the absolute value function. stands for "is in the set" so the statements should be read as saying that is in the set described in the two equivalent ways. stands for the union, so the statement that should be parsed as saying that or stands for set difference, so the statement can be parsed as saying that could be any value in except . The point is excluded because we do not want the value of at to affect the limit notion.
we have
The symbol stands for the absolute value function. stands for "is in the set" so the statement should be read as saying that is in the set .
The limit (also called the two-sided limit) is defined as a value such that . By the uniqueness theorem for limits, there is at most one value of for which . Hence, it makes sense to talk of the limit when it exists.
0:02:45.659,0:02:53.659
Vipul: [|x-c|] should be not equal to zero so zero
less than, exclude the point c itself,
0:02:54.810,0:02:56.930
less than delta. What do we have?
0:02:56.930,0:02:59.459
Rui: We have y is within.
0:02:59.459,0:03:04.260
Vipul: Well y is just f(x).
0:03:04.260,0:03:10.290
Rui: f(x_0)
0:03:14.290,0:03:16.819
Vipul: Well f(x) minus the claimed limit is?
0:03:17.219,0:03:18.040
Rui: L.
0:03:18.640,0:03:22.890
Vipul: You're thinking of continuity which is a
little different but here we have this less than?
0:03:22.890,0:03:24.569
Rui: Epsilon.
0:03:24.569,0:03:37.569
Vipul: Epsilon. Let me now just re-write these
conditions in interval notation.
0:03:37.830,0:03:40.031
What is this saying x in what interval? [ANSWER!]
0:03:40.040,0:03:43.519
Rui: c +- ...
0:03:43.519,0:03:49.840
Vipul: c- delta to c + delta excluding the
point c itself, that is what 0 < [|x -- c|] is telling us.
0:03:49.840,0:03:56.530
It is telling us x is within delta distance
of c, but it is not including c.
0:03:56.530,0:04:10.530
Another way of writing this is (c -- delta,c) union (c, c + delta)
0:04:12.810,0:04:19.340
x is either on immediate delta left of c or
it's on the immediate delta right of c.
0:04:21.040,0:04:31.040
You do something similar on the f(x) side
so what interval is this saying, f(x) is in what? [ANSWER!]
0:04:31.720,0:04:35.930
Rui: L -- epsilon, L + epsilon.
0:04:35.930,0:04:42.930
Vipul: Awesome. Instead of writing the conditions
in this inequality form you could have written
0:04:43.919,0:04:47.590
them in this form, so instead of writing this
you could have written this or this, instead
0:04:47.590,0:04:49.580
of writing this you could have written this.
0:04:50.080,0:04:59.500
If this statement is true, the way you read this is you say
limit as x approaches c of f(x) equals L.
0:04:59.500,0:05:07.500
Okay. Now how do we define the limit?
0:05:11.169,0:05:18.169
It's the number L for which the above holds. This should be
in quotes.
0:05:22.009,0:05:29.009
If a number L exists for which.
0:05:34.220,0:05:41.220
Now what would you need in order to show that
this definition makes sense?
0:05:47.919,0:05:52.919
Rui: I don't think I understand your question.
0:06:03.090,0:06:09.090
Vipul: What I mean is, what I wanted to ask
was what would you need to prove in order
0:06:09.990,0:06:14.889
to say the notion of the limit makes sense? Well,
you need to show that there is uniqueness here.
0:06:14.740,0:06:19.080
It cannot happen that the limit is some number
L and the limit is another number M so you
0:06:19.080,0:06:20.539
need to show uniqueness.
0:06:20.539,0:06:27.330
You need to show that if this holds for one
number L it cannot also hold for another number.
0:06:27.330,0:06:32.050
Once you have shown that then it you could
define it like this.
0:06:32.050,0:06:38.440
Now I should say "if it exists."
0:06:38.440,0:06:42.120
What I'm saying is that there is a uniqueness
theorem which we will prove some other time.
0:06:42.120,0:06:49.120
Which says that if this is true for one number
it cannot be true for any other number so
0:06:49.440,0:06:54.740
this statement is true for at the most one
value of L and if there is such a value of
0:06:54.740,0:06:55.050
L that's called the limit.
Checkpoint questions:
In order to make sense of where must the function be defined? Must be defined at ? If exists, what can we say about its value?
What's the formal definition of limit, i.e., what does mean?
How would you write the formal definition of limit using intervals rather than absolute value inequalities to describe where and should be?
Why is there a "" in the inequality in the definition? Why doesn't a appear in the part of the definition?
In order to be able to talk of the limit , what additional fact do we need beyond the definition of what means?
Left hand limit
Suppose is a function of one variable and is a point such that is defined on the immediate left of (note that may or may not be defined at ). In other words, there exists some value such that is defined on .
For a given value , we say that:
if the following holds:
For every , there exists such that for all satisfying , we have .
The definition is broken down into its four clauses below:
Clause
Interval description
Symbol explanations
For every
The symbol is a Greek lowercase letter pronounced "epsilon". Although the definition customarily uses the letter , it can be replaced by any other letter, as long as the letter is different from the other letters in use. The reason for sticking to a standard letter choice is that it reduces cognitive overload.
there exists such that
The symbol is a Greek lowercase letter pronounced "delta" Although the definition customarily uses the letter , it can be replaced by any other letter, as long as the letter is different from the other letters in use. The reason for sticking to a standard letter choice is that it reduces cognitive overload.
for all satisfying
The symbol stands for the absolute value function. stands for "is in the set" so the statements should be read as saying that is in the set describing the immediate -left of .
we have
The symbol stands for the absolute value function. stands for "is in the set" so the statement should be read as saying that is in the set .
The left hand limit (acronym LHL) is defined as a value such that . By the uniqueness theorem for limits (one-sided version), there is at most one value of for which . Hence, it makes sense to talk of the left hand limit when it exists.
Right hand limit
Suppose is a function of one variable and is a point such that is defined on the immediate right of (note that may or may not be defined at ). In other words, there exists some value such that is defined on .
For a given value , we say that:
if the following holds:
For every , there exists such that for all satisfying , we have .
The definition is broken down into its four clauses below:
Clause
Interval description
Symbol explanations
For every
The symbol is a Greek lowercase letter pronounced "epsilon". Although the definition customarily uses the letter , it can be replaced by any other letter, as long as the letter is different from the other letters in use. The reason for sticking to a standard letter choice is that it reduces cognitive overload.
there exists such that
The symbol is a Greek lowercase letter pronounced "delta" Although the definition customarily uses the letter , it can be replaced by any other letter, as long as the letter is different from the other letters in use. The reason for sticking to a standard letter choice is that it reduces cognitive overload.
for all satisfying
The symbol stands for the absolute value function. stands for "is in the set" so the statements should be read as saying that is in the set describing the immediate -right of .
we have
The symbol stands for the absolute value function. stands for "is in the set" so the statement should be read as saying that is in the set .
The right hand limit (acronym RHL) is defined as a value such that . By the uniqueness theorem for limits (one-sided version), there is at most one value of for which . Hence, it makes sense to talk of the right hand limit when it exists.
Side-by-side comparison of the definitions
Clause for two-sided limit
Clause for left hand limit
Clause for right hand limit
Comments
For every
For every
For every
identical so far
there exists such that
there exists such that
there exists such that
still identical
for all satisfying , i.e.,
for all satisfying , i.e.,
for all satisfying , i.e.,
this is the part that differs, in so far as it is the direction of domain approach that differs between the definitions.
we have , i.e.,
we have , i.e.,
we have , i.e.,
this part is again identical. Note that the left versus right is only about the direction of approach in the domain, not about the direction of approach of the function value.
0:00:15.940,0:00:20.740
Vipul: In this talk, I'm going to give definitions
of one-sided limits.
0:00:20.740,0:00:25.650
So it is going to be the left hand limit and
the right hand limit, and I'm going to basically
0:00:25.650,0:00:42.650
compare it with the definition of two-sided limit which was in
a previous video. Let's just write this down--left-hand limit.
0:00:44.110,0:00:48.679
Let me first remind you what the definition
of two-sided limit says.
0:00:48.679,0:00:57.679
So here's what it says. It says limit as x approaches
c, f(x) = L
0:00:58.469,0:01:03.140
so f has to be defined on the immediate left and
the immediate right of c.
0:01:03.140,0:01:07.960
It says that this is true if the following
holds so for every epsilon greater than zero
0:01:07.960,0:01:13.960
there exists a delta > 0 such that for all
x which are within delta of c
0:01:14.000,0:01:22.771
either delta on the left of c or within a delta on the
right of c we have that f(x) is within an epsilon
0:01:23.650,0:01:30.530
distance of L.
Okay. Now with the left and right hand limit
0:01:30.530,0:01:37.460
what we are trying to do we are trying to
consider only one-sided approaches on the, on the x
0:01:39.000,0:01:41.510
What will change when we do the left-hand limit,
0:01:42.001,0:01:44.641
what will be different from this definition?
[ANSWER!]
0:01:45.710,0:01:48.330
Rui: We approach c from the left.
0:01:48.330,0:01:52.790
Vipul: We'll approach c from the left so
what part of this definition will change? [ANSWER!]
0:01:52.790,0:01:54.880
Rui: From the fourth line?
0:01:54.880,0:01:56.890
Vipul: You mean this line?
0:01:56.890,0:02:06.810
Rui: Oh for all x within c distance, within delta distance of c
0:02:06.810,0:02:08.700
Vipul: So what will change?
0:02:08.700,0:02:14.020
Rui: We will not have (c, c + delta).
0:02:14.020,0:02:18.390
Vipul: This part won’t be there. We will
just be concerned about whether when x is
0:02:18.390,0:02:23.000
delta close on the left side of c, f(x) is here...
0:02:23.000,0:02:28.000
Will we change this one also to only include the left? [ANSWER!]
0:02:28.000,0:02:30.000
Or this one will remain as it is?
0:02:30.300,0:02:31.500
Rui: I think it will remain.
0:02:31.500,0:02:33.460
Vipul: It will remain as it is because we
0:02:33.460,0:02:35.340
are just saying as x approaches c from the left
0:02:35.340,0:02:36.340
f(x) approaches L.
0:02:36.340,0:02:43.340
We are not claiming that f(x) approaches L
from the left, okay? Let me make a number line picture.
0:02:51.750,0:02:56.130
We will do a full geometric understanding
of the thing later. Right now it's just very [formal].
0:02:56.130,0:03:00.850
So the function is defined on the immediate left
of c, maybe not defined at c. It is defined
0:03:00.850,0:03:01.920
on the immediate left of c.
0:03:01.920,0:03:06.410
We don’t even know if the function
is defined on the right of c and what we are
0:03:06.410,0:03:13.410
saying is that for any epsilon, so any epsilon
around L you can find a delta such that if you restrict
0:03:13.800,0:03:20.800
attention to the interval from c minus delta
to c [i.e., (c- delta, c) in math notation]
0:03:21.450,0:03:23.130
then the f value there is within the epsilon distance of L.
0:03:24.130,0:03:28.959
Now the f value could be epsilon to the left
or the right so we take left hand limit on
0:03:28.959,0:03:33.840
the domain side it doesn’t have to approach
from the left on the other side.
0:03:33.840,0:03:40.690
Let me just write down the full definition. We want to keep this on the side.
0:03:40.690,0:04:03.690
What it says that for every epsilon > 0 there
exists
0:04:05.180,0:04:16.680
by the way, the understanding of the what this definition
really means will come in another video you may have seen before this or after this
0:04:16.680,0:04:21.209
... for all x ... [continuing definition]
0:04:21.209,0:04:26.500
Now we should also change it if we are writing
in this form so how will it read now?
0:04:26.500,0:04:28.030
Rui: For all x ...
0:04:35.000,0:04:38.000
Vipul: So will you put x – c or c – x? [ANSWER!]
0:04:38.330,0:04:40.990
Rui: It will be x – c, oh c – x.
0:04:41.000,0:04:46.760
Vipul: c – x. Because you want c to be bigger
than x. You want x to be on the left of c.
0:04:46.850,0:05:01.850
What would this read, i.e. x is in (c – delta,c).
Okay.
0:05:05.000,0:05:11.460
What do we have? We have the same thing. This part doesn’t change.
0:05:13.000,0:05:19.000
Rui: f(x) is within epsilon distance of L.
0:05:34.400,0:05:40.400
Vipul: Why do I keep saying this thing about the
L approach doesn’t have to be from the left?
0:05:41.000,0:05:44.350
What’s the significance of that? Why is that important?
[ANSWER!]
0:05:45.000,0:05:51.000
Rui: It’s important because we don’t know
whether the function is decreasing or increasing
0:05:51.620,0:05:52.370
at that point.
0:05:52.370,0:05:55.750
Vipul: Yeah, so if your function is actually
increasing than L will also be approached
0:05:55.750,0:06:01.590
from the left, and if it’s decreasing it
will be approached from the right, but sometimes
0:06:01.590,0:06:07.590
it’s neither increasing nor decreasing, but it's still
true it approaches from one side, so that’s a
little complicated but the way
0:06:07.590,0:06:12.150
this comes up is that when you are dealing
with composition of functions, so when you
0:06:12.150,0:06:16.710
are doing one function and then applying another function to that and you have some results
0:06:16.710,0:06:18.440
with one-sided limits.
0:06:18.440,0:06:30.440
Let me just write this down. If you have one-sided
limits and you have composition,
0:06:31.610,0:06:39.550
so you are doing one function and then doing another
you have to be very careful.
0:06:45.050,0:06:48.350
You need to be very careful when you are doing
one-sided limits and composition.
0:06:48.360,0:06:57.360
Why? Because if you have g of f(x) and x approaches
to c from the left, f(x) approaches L but
0:06:57.850,0:06:59.280
not necessarily from the left.
0:06:59.280,0:07:03.560
You then you have another thing which is as
f(x) approaches L from the left, g of that
0:07:03.560,0:07:09.280
approaches something you just need to be careful
that when you compose things the sidedness
0:07:09.280,0:07:10.930
could change each time you compose.
0:07:10.930,0:07:14.590
Rui: Can you write a composition of the function
out?
0:07:14.590,0:07:17.870
Vipul: Not in this video. We will do that
in another video.
0:07:17.870,0:07:23.800
That’s something we will see in a subsequent
video but this is just something to keep in
0:07:23.800,0:07:27.770
mind so when you see that it will ring a bell.
0:07:30.770,0:07:31.880
Let us do... what other side is left? [pun unintended!]
Rui: Right?
Vipul: Right!
0:07:31.880,0:07:36.690
Vipul: By the way, you probably already know
this if you have seen limits intuitively so
0:07:36.690,0:07:42.300
I'm not stressing this too much but left hand
limit is really the limit as you approach
0:07:42.300,0:07:49.300
from the left. You are not moving toward the
left you are moving from the left to the point.
0:07:50.160,0:07:55.940
Right hand limit will be approach from the
right to the point so it is right, moving from
0:07:55.940,0:07:59.330
the right, so the words left and right are
describing where the limit is coming *from*,
0:07:59.330,0:08:06.330
not the direction which it is going to.
0:08:12.569,0:08:17.650
Now you can just tell me what will be the
corresponding thing. To make sense of this
0:08:17.650,0:08:19.819
notion we need f to be defined where? [ANSWER!]
0:08:19.819,0:08:21.699
Rui: On its right.
0:08:21.699,0:08:29.199
Vipul: On the immediate right of c. If it
is not defined on the immediate right it doesn’t
0:08:29.389,0:08:36.389
even make sense to ask this question what
the right hand limit is.
0:08:37.280,0:08:38.550
How will that be defined?
0:08:38.550,0:08:44.240
Rui: For every epsilon greater than zero
0:08:44.240,0:08:51.240
Vipul: The epsilon is the interval on which
you are trying to trap the function value.
0:08:51.500,0:08:54.279
Rui: There exists epsilon
0:08:54.279,0:08:55.890
Vipul: No, delta
0:08:55.890,0:09:14.890
Rui: delta> 0 such that for all x
with x – c > 0
0:09:15.040,0:09:22.040
Vipul: The general one is for all x with 0<|x-c|<delta
because you want to capture both the intervals.
0:09:23.170,0:09:29.270
In this one, the left hand limit one, we just
captured the left side interval.
0:09:29.270,0:09:39.270
Now in the right one we just want to capture
the right side interval, so as you said 0< x- c < delta.
0:09:44.180,0:09:51.480
In the picture, the function is defined, say c
to c + t and you are really saying you can
0:09:52.290,0:10:00.290
find delta if x is in here [between c and c + delta] which
actually... this is not including c, it is all the points
0:10:00.390,0:10:05.390
in the immediate right of c. We have? [ANSWER!]
0:10:06.000,0:10:13.000
Rui: The absolute value of f(x) – L is less
than epsilon.
0:10:20.010,0:10:22.010
Vipul: So f(x) is? Are we here? We have everything?
0:10:23.010,0:10:23.260
Rui: Yes.
0:10:26.190,0:10:30.890
Vipul: We have both of these here? So do you
see what’s the main difference between these
0:10:30.890,0:10:37.430
two and the actual [two-sided limit] definition?
0:10:37.430,0:10:42.930
For every epsilon there exists delta... the
first second and fourth line remain the same.
0:10:42.930,0:10:47.440
It is this line where you are specifying where
the x are that’s different.
0:10:47.440,0:10:53.000
In the two-sided thing the x could be either place.
0:10:53.300,0:10:55.200
For the left hand limit the x,
0:10:55.720,0:10:59.000
you just want x here [in (c - delta, c)] and
0:10:59.000,0:11:07.000
for the right hand limit you just want x in (c,c + delta).
0:11:07.000,0:11:09.000
Okay? [END!]
Checkpoint questions:
In order to make sense of , where must the function be defined? Must be defined at ? If exists, what can we say about its value?
The definitions of left hand limit, right hand limit and ordinary (two-sided) limit are pretty similar. There is only one clause that differs across the three definitions. What clause is this, and how does it differ across the definitions? Explain both in inequality notation and in interval notation.
Why should we be careful when dealing with one-sided limits in the context of function compositions?
Relation between the limit notions
The two-sided limit exists if and only if (both the left hand limit and right hand limit exist) and (they are equal to each other).
Explicitly, exists if all three of these conditions hold:
exists.
exists.
.
Moreover, in the event that both one-sided limits exist and are equal, the two-sided limit is equal to both of them.
Further, a particular value of works for a particular value of in the two-sided limit definition if and only if it works in both the left hand limit definition and the right hand limit definition.
Definition of finite limit for function of one variable in terms of a game
The formal definitions of limit, as well as of one-sided limit, can be reframed in terms of a game. This is a special instance of an approach that turns any statement with existential and universal quantifiers into a game.
Two-sided limit
Consider the limit statement, with specified numerical values of and and a specified function :
Note that there is one trivial sense in which the above statement can be false, or rather, meaningless, namely, that is not defined on the immediate left or immediate right of . In that case, the limit statement above is false, but moreover, it is meaningless to even consider the notion of limit. We therefore omit this sense from consideration and consider instead only the situation where is defined on the immediate left and immediate right of .
The game is between two players, a Prover whose goal is to prove that the limit statement is true, and a Skeptic (also called a Verifier or sometimes a Disprover) whose goal is to show that the statement is false. The game has three moves:
First, the skeptic chooses , or equivalently, chooses the target interval in which the skeptic is challenging the prover to trap the function.
Then, the prover chooses , or equivalently, chooses the interval .
Then, the skeptic chooses a value satisfying , or equivalently, , which is the same as .
Now, if (i.e., ), the prover wins. Otherwise, the skeptic wins.
We say that the limit statement
is true if the prover has a winning strategy for this game. The winning strategy for the prover basically constitutes a strategy to choose an appropriate in terms of the chosen by the skeptic. Thus, it is an expression of as a function of . Verbally, the goal of the prover is to choose a value of so that when the input is restricted to being within distance of , the output is trapped to within distance of the claimed limit .
We say that the limit statement
is false if the skeptic has a winning strategy for this game. The winning strategy for the skeptic involves a choice of , and a strategy that chooses a value of (constrained in the specified interval) based on the prover's choice of .
Let's review the definition in conjunction with the game along with a deeper semantic understanding of the steps:
Step no.
Clause of definition
Who moves?
What is chosen?
Constraints on the choice
Comment
1
For every
Skeptic
Must be positive
The "for every" corresponds to the idea that the move is made by the side that does not have a winning strategy, because we want to argue that the side that does have a winning strategy can win no matter what.
2
there exists such that
Prover
Must be positive
The "there exists" corresponds to the idea that the move is made by the side that has a winning strategy, because that side gets to choose a favorable value of the variable (in this case ).
3
for all satisfying ,
Skeptic
Must be within the interval
The "for all" corresponds to the idea that the move is made by the side that does not have a winning strategy, because we want to argue that the side that does have a winning strategy can win no matter what.
4
we have
Neither; it's time for the judge to decide
--
If (the condition that we desire) the prover wins. Else, the skeptic wins.
Slight subtlety regarding domain of definition: [SHOW MORE]
The domain of definition issue leads to a couple of minor subtleties:
A priori, it is possible that the chosen by the skeptic is outside the domain of , so it does not make sense to evaluate . In the definition given above, this would lead to the game being won by the skeptic. In particular, if is not defined on the immediate left or right of , the skeptic can always win by picking outside the domain.
It may make sense to restrict discussion to the cases where is defined on the immediate left or right of . Explicitly, we assume that is defined on the immediate left and immediate right, i.e., there exists such that is defined on the interval . In this case, it does not matter what rule we set regarding the case that the skeptic picks outside the domain. To simplify matters, we could alter the rules in any one of the following ways, and the meaning of limit would remain the same as in the original definition:
We could require (as part of the game rules) that the prover pick such that . This pre-empts the problem of picking -values outside the domain.
We could require (as part of the game rules) that the skeptic pick in the domain, i.e., pick with and .
We could alter the rule so that if the skeptic picks outside the domain, the prover wins (instead of the skeptic winning).
0:08:35.050,0:08:38.669
Vipul: Yeah. True if the prover has a winning strategy.
0:08:38.669,0:08:44.910
It is not just enough to say that the prover
won the game some day but the prover should
0:08:44.910,0:08:50.220
be able to win the game regardless of how
smart the skeptic is or regardless of how
0:08:50.220,0:08:53.960
experienced the skeptic is or regardless of
how the skeptic plays.
0:08:53.960,0:09:00.960
That's why all the moves of the skeptic
are prefaced with a "for every." Right?
0:09:02.230,0:09:07.560
Whereas all the moves of the prover are prefaced,
(well there is only one move really of the
0:09:07.560,0:09:11.180
prover) are prefaced with "there exists"
because the prover controls his own choices.
0:09:11.180,0:09:15.360
When it is the prover's turn it's enough
to say "there exists" but since the prover doesn't
0:09:15.360,0:09:21.590
control what the skeptic does all the skeptic
moves are prefaced with "for every."
0:09:21.590,0:09:26.150
By the way, there is a mathematical notation
for these things.
0:09:26.150,0:09:31.730
There are mathematical symbols for these,
which I'm not introducing in this video,
0:09:31.730,0:09:37.920
but if you have seen them and got confused
then you can look at the future video where
0:09:37.920,0:09:40.500
I explain the mathematical symbols.
Negation of limit statement and non-existence of limit
We now consider the explicit description of the definition for the case that the skeptic has a winning strategy for the limit game for , i.e., for the limit statement being false.
In words, the definition is:
There exists such that for every , there exists satisfying and .
Let's review the definition in conjunction with the game along with a deeper semantic understanding of the steps:
Step no.
Clause of definition for original limit statement (i.e., prover has a winning strategy)
Clause of definition for skeptic having a winning strategy
Who moves?
What is chosen?
Constraints on the choice
Comment
1
For every
There exists such that
Skeptic
Must be positive
Whether we use "for every" or "there exists" depends on who we're rooting for.
2
there exists such that
for every ,
Prover
Must be positive
Whether we use "for every" or "there exists" depends on who we're rooting for.
3
for all satisfying ,
there exists satisfying and
Skeptic
Must be within the interval
Whether we use "for every" or "there exists" depends on who we're rooting for.
0:01:26.720,0:01:33.720
Ok, so in this talk, we are going to give the definition
of what it means to say that this statement,
0:01:34.250,0:01:37.940
the one up here, is false.
0:01:37.940,0:01:41.300
So far we've looked at what it means for this
statement to be true.
0:01:41.300,0:01:44.960
Now we are going to look at what it means
for the statement to be false.
0:01:44.960,0:01:48.340
Basically, you just use the same definition,
but you would change a little bit of what
0:01:48.340,0:01:49.490
it looks like.
0:01:49.490,0:01:54.130
Let me first remind you of the limit game
because that is a very nice way of thinking
0:01:54.130,0:01:57.380
about what it means to be true and false.
0:01:57.380,0:01:58.860
What does the limit game say?
0:01:58.860,0:02:01.680
It is a game between two players, a prover
and a skeptic.
0:02:01.680,0:02:04.680
What is the goal of the prover? [ANSWER!]
0:02:04.680,0:02:06.310
Rui: To show he is right.
0:02:06.310,0:02:07.930
Vipul: To show that this is true.
0:02:07.930,0:02:08.489
Rui: True.
0:02:08.489,0:02:12.830
Vipul: The skeptic is trying to show that
this is false, or at least trying to come
0:02:12.830,0:02:16.730
up with the strongest evidence to suggest
that this is false.
0:02:16.730,0:02:18.090
How does the game proceed?
0:02:18.090,0:02:23.349
The skeptic begins by choosing an epsilon
greater than zero.
0:02:23.349,0:02:25.200
What is the skeptic effectively trying to
pick?
0:02:25.200,0:02:30.769
The skeptic is effectively trying to pick
this neighborhood of L and trying to challenge
0:02:30.769,0:02:36.579
the prover to trap the function value for
x within that neighborhood.
0:02:36.579,0:02:40.719
What's that neighborhood the skeptic is
secretly picking? [ANSWER!]
0:02:40.719,0:02:43.909
Rui: L -- epsilon [to L + epsilon]
0:02:43.909,0:02:50.909
Vipul: Ok, the prover chooses a delta greater
than zero so the prover is now basically trying
0:02:53.040,0:03:00.040
to pick a neighborhood of c, the point near the
domain points, and
0:03:02.650,0:03:09.650
then the skeptic will then pick a value x, which is within the interval delta distance
of c except the point c itself.
0:03:10.120,0:03:16.200
That's either delta interval on the left
or delta interval on the right of c.
0:03:16.200,0:03:20.569
Then the judge comes along and computes this
value, absolute value f(x) minus...Are we,
0:03:20.569,0:03:21.739
is this in the picture?
0:03:21.739,0:03:22.700
Rui: Yes.
0:03:22.700,0:03:27.329
Vipul: If it is less than epsilon then the
prover would have won, but now we want to
0:03:27.329,0:03:34.329
see if the skeptic wins if it is greater or
equal to epsilon, that means f(x) is not in
0:03:35.569,0:03:36.129
the epsilon...
0:03:36.129,0:03:37.249
Rui: Neighborhood.
0:03:37.249,0:03:42.459
Vipul: This video assumes you have already
seen the previous videos where we give these
0:03:42.459,0:03:48.689
definitions and so I'm sort of reviewing it
quickly, but not explaining it in full detail.
0:03:48.689,0:03:54.069
So, the skeptic wins if f(x) is outside this
interval, that means the prover failed to
0:03:54.069,0:03:58.069
rise to the skeptic's challenge of trapping
the function.
0:03:58.069,0:04:05.069
Let's now try to work out concretely what
the definition would read.
0:04:06.590,0:04:10.439
The skeptic is the one in control because
you want to figure out whether the skeptic
0:04:10.439,0:04:12.639
has a winning strategy.
0:04:12.639,0:04:17.690
Ok, so let me just say this clearly, this
is just saying when does the skeptic win?
0:04:17.690,0:04:21.090
Now in order to say this limit statement is
false, we need something stronger. What do
0:04:21.090,0:04:25.360
we need to say this is false? [ANSWER!]
0:04:25.360,0:04:26.450
The skeptic should have...
0:04:26.450,0:04:28.820
Rui: Should have a winning strategy.
0:04:28.820,0:04:30.410
Vipul: A winning strategy.
0:04:30.410,0:04:34.229
The skeptic should have a strategy so that
whatever the prover does, the skeptic has
0:04:34.229,0:04:36.139
some way of winning.
0:04:36.139,0:04:41.229
What should this read...if you actually translate
it to the definition?
0:04:41.229,0:04:44.169
Rui: There exists an...
0:04:44.169,0:04:46.000
Vipul: There exists epsilon
0:04:46.000,0:04:51.000
Rui: ...an epsilon greater than zero.
0:04:58.000,0:05:00.000
Vipul: Okay. Such that...
0:05:00.280,0:05:07.210
Rui: For every delta greater than zero.
0:05:07.210,0:05:10.870
Vipul: So the skeptic, when it's the skeptic's
move the skeptic says "there exists."
0:05:10.870,0:05:14.310
If anything works, the skeptic can pick that,
but when it's the provers move, the skeptic
0:05:14.310,0:05:15.699
has no control.
0:05:15.699,0:05:30.699
This should read, for every delta greater
than zero...What will the next part read?
0:05:31.770,0:05:33.930
Rui: There exists an x.
0:05:33.930,0:05:40.930
Vipul: Exists x in this interval.
0:05:44.289,0:05:45.340
Rui: Yeah.
0:05:45.340,0:05:50.159
Vipul: Which you often see it written in a
slightly different form.
0:05:50.159,0:05:57.159
Maybe, I don't have space here, so here
it is also written as "0 ...", are we down here?
0:05:59.960,0:06:01.560
Rui: Yes.
0:06:01.560,0:06:04.470
Vipul: This is the form it's usually written in
concise definitions.
0:06:04.470,0:06:20.710
We have this...So the definition, maybe it's not
clear, but the definition would read like that.
0:06:20.710,0:06:25.419
So there exists Epsilon greater than zero such
that for every delta greater than zero there
0:06:25.419,0:06:30.879
exists x, in here, which you could also write
like this, such that, I guess I should put
0:06:30.879,0:06:35.310
the "such that." [writes it down]
0:06:35.310,0:06:39.849
Such that. absolute value of f(x) -- L is greater
than or equal to epsilon
0:06:39.849,0:06:44.680
Let me just compare it with the usual definition
for the limit to exist.
0:06:44.680,0:06:47.750
The colors are in a reverse chrome.
0:06:47.750,0:06:52.860
That's fine. For every epsilon greater than
zero became there exists epsilon greater than
0:06:52.860,0:06:55.879
zero because the player who is in control
has changed.
0:06:55.879,0:06:59.789
There exists delta greater than zero became
for every delta greater than zero, for all
0:06:59.789,0:07:05.139
x with this became their exists x satisfying
this condition.
0:07:05.139,0:07:07.629
What happened to the last clause?
0:07:07.629,0:07:12.099
The less than Epsilon begin greater than or
equal to.
0:07:12.099,0:07:17.069
The last clause just got reversed in meaning,
all the others, we just changed the quantifier
0:07:17.069,0:07:22.389
from "for all" to "there exists" and from "there
exists" to "for all" and that is just because
0:07:22.389,0:07:25.770
we changed who is winning.
0:07:25.770,0:07:30.439
If you have seen some logic, if you ever see
logic, then there are some general rules of
0:07:30.439,0:07:33.650
logic as to how to convert a statement to
its opposite statement.
0:07:33.650,0:07:38.610
This is a general rule that "for all" becomes
"there exists" and "there exists" becomes "for all."
Non-existence of limit
The statement does not exist could mean one of two things:
is not defined around , i.e., there is no for which is defined on . In this case, it does not even make sense to try taking a limit.
is defined around , except possibly at , i.e., there is for which is defined on . So, it does make sense to try taking a limit. However, the limit still does not exist.
The formulation of the latter case is as follows:
For every , there exists such that for every , there exists satisfying and such that .
We can think of this in terms of a slight modification of the limit game, where, in our modification, there is an extra initial move by the prover to propose a value for the limit. The limit does not exist if the skeptic has a winning strategy for this modified game.
An example of a function that does not have a limit at a specific point is the sine of reciprocal function. Explicitly, the limit:
does not exist. The skeptic's winning strategy is as follows: regardless of the chosen by the prover, pick a fixed (independent of , so can be decided in advance of the game -- note that the skeptic could even pick and the strategy would still work). After the prover has chosen a value , find a value such that the function value lies outside . This is possible because the interval has width , hence cannot cover the entire interval , which has width 2. However, the range of the function on is all of .
Crucially, the inability of the prover to trap the function value close to any point as is the reason the limit fails to exist.
0:00:31.170,0:00:38.170
Vipul: Ok, so this talk is going to be about
why under certain circumstances limits don't exist
0:00:39.800,0:00:46.800
We are going to take this example of a function
which is defined like this: sin of one over x
0:00:47.699,0:00:51.360
Obviously, that definition doesn't work
when x equals zero.
0:00:51.360,0:00:57.260
So this is a function defined only for all non-zero
reals.
0:00:57.260,0:01:01.050
The goal is to figure out what the limit as
x approaches 0 of f(x) is.
0:01:01.050,0:01:06.630
Here is a graph of the function. This is a
y axis, and x axis.
0:01:06.630,0:01:08.490
The function looks like this.
0:01:08.490,0:01:10.680
It is oscillatory.
0:01:10.680,0:01:16.270
As you approach zero it oscillates more, faster
and faster.
0:01:16.270,0:01:19.070
What are the upper and lower limits of oscillation?
0:01:19.070,0:01:25.580
Actually all these things should be the same
height.
0:01:25.580,0:01:29.760
My drawing wasn't good, but, it should all
be the same height, above and below.
0:01:29.760,0:01:31.290
What are these upper and lower limits? [ANSWER!]
0:01:31.290,0:01:32.790
Rui: 1 and -1.
0:01:32.790,0:01:39.790
Vipul: So the lower limit is negative one
and the upper limit is one. Ok, good.
0:01:39.829,0:01:46.829
So what does it mean, what is the limit at
zero for this function? [ANSWER!]
0:01:46.850,0:01:53.850
This is where...you need to really think, so
I might say ok the limit is, looks like it's zero.
0:01:58.259,0:01:58.509
0:01:58.469,0:02:04.749
At zero, you say that looks neat, that looks
right because you see when the x value approaches,
0:02:04.749,0:02:09.190
comes close to zero, the f(x) value also comes
close to zero.
0:02:09.190,0:02:12.700
It keeps oscillating between -1and 1,
and it keeps coming.
0:02:12.700,0:02:19.700
I draw a very small ball around zero, like
that.
0:02:19.780,0:02:22.700
The function is going to keep entering this
ball.
0:02:22.700,0:02:27.060
A ball or a square one or whatever.
0:02:27.060,0:02:34.060
A very small neighborhood of this origin point
here in this two-dimensional picture.
0:02:35.230,0:02:40.459
The function graph is going to enter that
repeatedly.
0:02:40.459,0:02:42.010
Do you think the limit is zero? [ANSWER!]
0:02:42.010,0:02:42.830
Rui: No.
0:02:42.830,0:02:46.860
Vipul: No? Why not? Isn't it coming really
close to zero?
0:02:46.860,0:02:47.430
Rui: Sometimes.
0:02:47.430,0:02:49.140
Vipul: What do you mean "sometimes?"
0:02:49.140,0:02:56.140
Rui: It means sometimes it is real close to
zero and then it flies away.
0:02:56.870,0:03:03.870
Vipul: Ok, "flies away." [Hmm] So what's
your objection? What is not happening?
0:03:04.019,0:03:06.010
Rui: We can not trap.
0:03:06.010,0:03:07.239
Vipul: We cannot trap...
0:03:07.239,0:03:11.909
Rui: ...trap it in a neighborhood of zero.
0:03:11.909,0:03:18.480
Vipul: Function not trapped.
0:03:18.480,0:03:20.110
What should the limit be if it is not zero?
0:03:20.110,0:03:24.849
Should it be half, two-thirds, what should
the limit be? [ANSWER!]
0:03:24.849,0:03:31.849
(I'll explain this later), what do you think
the limit should be?
0:03:34.659,0:03:36.730
Rui: It doesn't have a limit.
0:03:36.730,0:03:38.299
Vipul: It doesn't have a limit.
0:03:38.299,0:03:39.790
Ok, so what does that mean?
0:03:39.790,0:03:45.290
Whatever limit you claim the function has
you are wrong...If you claim the function had
0:03:45.290,0:03:49.170
any numerical limit, if you claim if it is half you
are wrong.
0:03:49.170,0:03:50.640
If you claim minus half you are wrong.
0:03:50.640,0:03:52.720
If you claim the limit is 50, you are wrong.
0:03:52.720,0:03:54.959
Whatever claim you make about the limit,
you are wrong.
0:03:54.959,0:04:00.780
So let's try to think of this in terms of the
game between a prover and a skeptic.
0:04:00.780,0:04:02.730
(You should go and review that video
0:04:02.730,0:04:09.730
or read the corresponding material to understand
what I am going to say.)
0:04:09.829,0:04:13.969
It's good if you have also seen the video
on the definition of limit statement being
0:04:13.969,0:04:17.709
false, which builds on that.
0:04:17.709,0:04:21.620
What I am now asking you, what does it mean
to say the limit does not exist?
0:04:21.620,0:04:23.980
As x approaches c [limit] of f(x) does not exist.
0:04:23.980,0:04:27.810
Here c is zero, but that is not relevant...
that is not necessary for the definition.
0:04:27.810,0:04:32.910
Well it is the usual way we say that the
limit statement is false except we need to
0:04:32.910,0:04:37.170
add one step in the beginning, which is for
every L in R [the reals].
0:04:37.170,0:04:42.460
It says that for every L in R [the reals] the statement
limit x approaches c, f(x) equals L, is false.
0:04:42.460,0:04:43.900
So how does it read?
0:04:43.900,0:04:48.220
It says, for every L in R [the reals] there exists epsilon
greater than zero such that for every delta
0:04:48.220,0:04:55.030
greater than zero there exists x, within the
delta neighborhood of c such that f(x) is
0:04:55.030,0:04:58.590
not in the epsilon neighborhood of L.
0:04:58.590,0:05:05.590
How would you interpret this in terms of a
game between a prover and a skeptic?[ANSWER, THINKING ALONG!]
0:05:06.470,0:05:11.570
Rui: For every limit the prover proposes...
0:05:11.570,0:05:16.420
Vipul: This is not quite the same as the limit
game which you may have seen in a previous
0:05:16.420,0:05:21.170
video which was assuming that the limit was
already given as a part of the game.
0:05:21.170,0:05:28.170
This is sort of a somewhat more general game or
a more meta game where part of the game
0:05:28.420,0:05:31.950
is also the prover trying to specify what
the limit should be.
0:05:31.950,0:05:37.100
The first step the prover plays, the prover
is in black, skeptic is in red.
0:05:37.100,0:05:43.290
The first step the prover plays, proposes
a value of the limit. Then?
0:05:43.290,0:05:47.280
Rui: The skeptic chooses an epsilon.
0:05:47.280,0:05:50.020
Vipul: What's the goal of the skeptic in choosing
the epsilon?
0:05:50.020,0:05:56.740
The goal of the skeptic is.. so let's say
the prover chose a limit value L here, that's
0:05:56.740,0:05:58.470
numerical value L here.
0:05:58.470,0:06:00.050
The skeptic picks epsilon.
0:06:00.050,0:06:06.650
The skeptic will pick epsilon, which means
the skeptic is picking this band from L minus
0:06:06.650,0:06:12.400
epsilon to L plus epsilon.
0:06:12.400,0:06:14.270
Now what does the prover try to do?
0:06:14.270,0:06:19.000
The prover tries to pick a delta. What is
the prover trying to do?
0:06:19.000,0:06:24.490
Find a neighborhood of c, such that the
function in that neighborhood of c the function
0:06:24.490,0:06:28.370
is trapped within epsilon of L.
0:06:28.370,0:06:32.740
So in our case, c is zero in this example,
so the prover will be trying to pick a neighborhood
0:06:32.740,0:06:39.740
of zero, is something like... zero plus delta
on the right and zero minus delta on the left.
0:06:44.620,0:06:45.750
What's the goal of the prover?
0:06:45.750,0:06:50.840
To say that whenever x is in this interval,
for all x,
0:06:50.840,0:06:53.500
The prover is trying to say that all for x
in here, the function [difference from L] is less than epsilon.
0:06:53.500,0:06:56.170
The skeptic who is trying to disprove that.
0:06:56.170,0:06:59.060
What does the skeptic need to do?
0:06:59.060,0:07:03.900
Rui: Every time the prover finds an x.
0:07:03.900,0:07:07.540
Vipul: Well the prover finds, picks the delta,
what does the skeptic try to do?
0:07:07.540,0:07:08.480
Rui: Just pick an x.
0:07:08.480,0:07:10.550
Vipul: Picks an x such that the function...
0:07:10.550,0:07:12.140
Rui: Is out of the...
0:07:12.140,0:07:13.960
Vipul: Is outside that thing.
0:07:13.960,0:07:24.960
Let me make this part a little bit more...so
here you have... the same colors.
0:07:25.150,0:07:41.150
This is
the axis...The skeptic...The prover has picked
this point and the skeptic has picked epsilon.
0:07:41.780,0:07:46.670
So this is L plus epsilon, L minus epsilon.
0:07:46.670,0:07:50.460
The prover is now, it so happens that c is
zero here.
0:07:50.460,0:07:56.690
So that everything is happening near the y
axis.
0:07:56.690,0:08:03.690
Now, the prover wants to pick a delta, the
prover wants to pick, like this, should be
0:08:07.320,0:08:07.910
the same.
0:08:07.910,0:08:14.910
So this is c plus delta which c is zero, so
zero plus delta and zero minus delta.
0:08:17.810,0:08:21.960
Now, under what conditions...What happens
next?
0:08:21.960,0:08:28.240
The prover is implicitly trying to claim that
the function, when the x value is close here,
0:08:28.240,0:08:30.520
the function value is trapped here.
0:08:30.520,0:08:35.089
What the skeptic wants to show is that, that's
not true.
0:08:35.089,0:08:39.830
If it isn't true, in order to do that, the
skeptic should pick a value of x.
0:08:39.830,0:08:46.830
So the skeptic needs to pick a value of x
somewhere in this interval such that at that
0:08:48.110,0:08:55.110
value of f(x)...let me just make the x axis...so
the skeptic wants to pick a value of x, maybe
0:08:59.209,0:09:06.209
its somewhere here, such that when you evaluate
the function at x it lies outside.
0:09:07.269,0:09:11.720
If when you evaluate the function at x, and it lies
outside this strip then the skeptic wins and
0:09:11.720,0:09:16.290
if the value of the function of x is inside
the strip then the prover wins.
0:09:16.290,0:09:23.290
Now looking back at this function, the question
is, can the prover pick an L such that regardless,
0:09:25.209,0:09:31.779
so can the prover pick a value of L such that...Is
this whole thing coming?
0:09:31.779,0:09:37.860
Such that regardless of the epsilon that the
skeptic picks, there exists a delta such that
0:09:37.860,0:09:44.439
for all x the function is trapped? Or is it
instead true that the skeptic will win? (i.e.) Is
0:09:44.439,0:09:50.579
it true that whatever L the prover picks there
exists an epsilon, since the skeptic picks
0:09:50.579,0:09:57.360
an epsilon, such that whatever delta the prover
picks the function in not in fact, trapped
0:09:57.360,0:10:00.399
here. What do you think looking at the picture
here?
0:10:00.399,0:10:05.329
Can you trap the function in a rectangle
like this? [ANSWER!]
0:10:05.329,0:10:06.100
Rui: No.
0:10:06.100,0:10:09.930
Vipul: Well, not if it is a very small rectangle.
0:10:09.930,0:10:16.930
What should the skeptic's strategy be?
0:10:17.060,0:10:23.930
The claim is that the limit does not exist,
that is the claim.
0:10:23.930,0:10:25.990
The claim is that this limit doesn't exist.
0:10:25.990,0:10:29.750
What is the skeptic's strategy?
0:10:29.750,0:10:31.990
What do you mean by skeptic strategy?
0:10:31.990,0:10:37.370
Well, the skeptic should have some strategy
that works, so the skeptic should pick an
0:10:37.370,0:10:43.290
epsilon that is smart and then the skeptic
should pick an x that works.
0:10:43.290,0:10:50.209
What epsilon should the skeptic pick? Suppose
the skeptic picks epsilon as 50 million,
0:10:50.209,0:10:52.050
is that a winning strategy?
0:10:52.050,0:10:52.790
Rui: No.
0:10:52.790,0:10:53.899
Vipul: Why not?
0:10:53.899,0:10:58.300
Rui: He should pick something between -1 and
1, right?
0:10:58.300,0:11:01.920
Vipul: Well epsilon is a positive number so
what do you mean?
0:11:01.920,0:11:04.600
Rui: Oh, anything between one, smaller.
0:11:04.600,0:11:05.230
Vipul: Smaller than...
0:11:05.230,0:11:08.999
Rui: Less than one. Epsilon.
0:11:08.999,0:11:12.470
Vipul: Less than one. Why will that work?
0:11:12.470,0:11:19.470
Rui: Because even if it is less than one then
anything, no matter what kind of delta...
0:11:20.930,0:11:27.930
Vipul: Whatever L the prover picked...What
is the width of this interval? The distance
0:11:28.209,0:11:29.589
from the top and the bottom is?
0:11:29.589,0:11:30.279
Rui: 2
0:11:30.279,0:11:30.980
Vipul: [2 times] epsilon.
0:11:30.980,0:11:31.680
Rui: [2 times] epsilon.
0:11:31.680,0:11:38.680
Vipul: 2 epsilon. If epsilon
is less than one, the skeptic's strategy is
pick epsilon less than one any epsilon.
0:11:43.089,0:11:50.089
The skeptic can fix epsilon in the beginning, maybe pick
epsilon as 0.1 or something, but any epsilon
0:11:50.610,0:11:52.019
less than one will do.
0:11:52.019,0:11:59.019
In fact epsilon equal to one will do. Let
us play safe and pick epsilon as 0.1.
0:11:59.810,0:12:00.999
Why does it work?
0:12:00.999,0:12:06.600
Because this 2 epsilon cannot include both
one and minus one.
0:12:06.600,0:12:12.649
It cannot cover this entire thing because
this has width two, from one to minus one.
0:12:12.649,0:12:17.589
If the skeptic picks an epsilon less than
one, regardless of the L the prover has tried,
0:12:17.589,0:12:23.079
the strip is not wide enough to include everything
from minus one to one.
0:12:23.079,0:12:27.990
Regardless of what Delta the prover picks,
we know that however small an interval we
0:12:27.990,0:12:32.180
pick around zero, the function is going to
take all values from negative one to one in
0:12:32.180,0:12:35.759
that small interval.
0:12:35.759,0:12:40.819
Now the skeptic will be able to find an x
such that the function value lies outside
0:12:40.819,0:12:42.290
the interval.
0:12:42.290,0:12:45.579
The skeptic should...the key idea is that
the skeptic pick epsilon small enough, in
0:12:45.579,0:12:50.360
this case the skeptic's choice of epsilon
doesn't depend on what L the prover chose.
0:12:50.360,0:12:51.269
It need not.
0:12:51.269,0:12:52.889
The strategy doesn't.
0:12:52.889,0:12:59.889
Then after the prover has picked a delta,
picked an x such that the function lies outside.
0:13:01.249,0:13:07.410
Regardless of the L the prover picks,
that L doesn't work as a limit because
0:13:07.410,0:13:10.550
the skeptic wins and so the limit doesn't
exist.
Strategic aspects
The strategy of small
In the game formulation of the limit, the following loose statements are true:
"Smaller is smarter" for the skeptic, i.e., the smaller the choice of , the better the outlook is for the skeptic to win.
"Smaller is smarter" for the prover, i.e., the smaller the choice of , the better the outlook is for the prover to win.
In other words, each side benefits by making the crucial move of that side as small as possible. However, there does not exist any single arbitrarily small number -- this is related to the observation in the motivation section that there is no such thing as a single arbitrarily close number. Thus, saying "choose as small a value as possible" is not a coherent strategy. What we can say is the following:
If a value of works for a given value of , the same value of also works for larger choices of .
If a value of works for a given value of , smaller values of also work for the same choice of .
Prover's strategy revisited
The prover, in choosing a winning strategy, must specify a rule that can determine a value of that works in terms of the value of specified by the skeptic. In other words, the prover must have a way of specifying as a function of .
The skeptic also chooses in the next move. However, the prover has no way of knowing the value of that the skeptic plans to pick. Thus, in order for the prover to have a winning strategy, the prover's choice of should be such that no matter what the skeptic picks, the prover wins.
Skeptic's strategy revisited
The skeptic, in choosing a winning strategy, must specify the value of and then specify how to pick a value of that works. When picking the value of , the skeptic does not know what the prover will pick. Thus, the skeptic's choice of cannot be dependent on the prover's subsequent choice of .
However, when picking the value of , the skeptic is aware of (and constrained by) the prover's choice of .
Misconceptions
Most misconceptions associated with the formal definition of limit have to do with the ordering of the moves in the game, who's in charge of what move, and what information each person has at the time of making the move. We describe some common misconceptions below.
0:09:36.080,0:09:39.529
KM: Then the skeptic has to choose the x value.
0:09:39.529,0:09:42.470
Vipul: x value. So, when the prover is deciding
the strategy, when the
0:09:42.470,0:09:45.860
prover is choosing the delta, what information
does the prover have?
0:09:45.860,0:09:48.410
KM: He just has the information on epsilon.
0:09:48.410,0:09:50.500
Vipul: Just the information on epsilon. So?
0:09:50.500,0:09:57.060
KM: So, in this case the mistake was that
because he didn't know the x value yet?
0:09:57.060,0:10:03.100
Vipul: The strategy cannot depend on x.
0:10:03.100,0:10:04.800
KM: Yeah.
0:10:04.800,0:10:09.790
Vipul: So, the prover is picking the
delta based on x but the
0:10:09.790,0:10:12.660
prover doesn't know x at this stage when
picking the delta. The delta
0:10:12.660,0:10:15.910
that the prover chooses has to be completely
a function of epsilon
0:10:15.910,0:10:19.680
alone, it cannot depend on the future moves
of the skeptic because the
0:10:19.680,0:10:23.700
prover cannot read the skeptic's mind. Okay?
And doesn't know what the
0:10:23.700,0:10:24.800
skeptic plans to do.
0:10:24.800,0:10:31.800
So that is the ... that's the proof. I call
this the ...
0:10:42.240,0:10:43.040
Can you see what I call this?
0:10:43.040,0:10:45.399
KM: The strongly telepathic prover.
0:10:45.399,0:10:51.470
Vipul: So, do you know what I meant by that?
Well, I meant the prover
0:10:51.470,0:10:58.470
is reading the skeptic's mind. All
right? It's called telepathy.
0:11:07.769,0:11:10.329
0:11:10.329,0:11:17.329
Okay, the next one.
0:11:25.589,0:11:30.230
This one says there's a function defined piecewise. Okay? It's defined
0:11:30.230,0:11:34.829
as g(x) is x when x is rational and zero when
x is irrational. So,
0:11:34.829,0:11:41.829
what would this look like? Well, pictorially, there's a line y
0:11:42.750,0:11:49.510
equals x and there's the x-axis and the
graph is just the irrational x
0:11:49.510,0:11:52.750
coordinate parts of this line and the rational
x coordinate parts of
0:11:52.750,0:11:56.350
this line. It's kind of like both these
lines but only parts of
0:11:56.350,0:11:58.529
them. Right?
0:11:58.529,0:12:02.079
Now we want to show that limit as x approaches
zero of g(x) is
0:12:02.079,0:12:06.899
zero. So just intuitively, do you think the statement
is true? As x goes
0:12:06.899,0:12:09.910
to zero, does this function go to zero?
0:12:09.910,0:12:10.610
KM: Yes.
0:12:10.610,0:12:17.610
Vipul: Because both the pieces are going to
zero. That's the intuition. Okay?
0:12:20.610,0:12:24.089
This is the proof we have here. So the idea
is we again think about it
0:12:24.089,0:12:27.790
in terms of the game. The skeptic first picks
the epsilon, okay? Now
0:12:27.790,0:12:30.779
the prover has to choose the delta, but
there are really two cases
0:12:30.779,0:12:35.200
on x, right? x rational and x irrational.
So the prover chooses the
0:12:35.200,0:12:39.459
delta based on whether the x is rational
or irrational, so if
0:12:39.459,0:12:43.880
the x is rational then the prover just picks
delta equals epsilon, and
0:12:43.880,0:12:48.339
that's good enough for rational x, right?
Because for rational x the
0:12:48.339,0:12:51.410
slope of the line is one so picking delta
as epsilon is good enough.
0:12:51.410,0:12:55.760
For irrational x, if the skeptic's planning
to choose an irrational x
0:12:55.760,0:12:59.730
then the prover can just choose any delta
actually. Like just fix
0:12:59.730,0:13:03.880
a delta in advance. Like delta is one or
something. Because if x is
0:13:03.880,0:13:10.430
irrational then it's like a constant function
and therefore, like, for
0:13:10.430,0:13:14.970
any delta the function is trapped within epsilon
distance of the claimed
0:13:14.970,0:13:16.970
limit zero. Okay?
0:13:16.970,0:13:19.950
So the prover makes two cases based
on whether the skeptic is going
0:13:19.950,0:13:26.950
to pick a rational or an irrational x
and based on that if
0:13:27.040,0:13:30.730
it's rational this is the prover's strategy,
if it's irrational then
0:13:30.730,0:13:34.050
the prover can just pick any delta.
0:13:34.050,0:13:37.630
Can you tell me what's wrong with this proof?
0:13:37.630,0:13:44.630
KM: So, he [the prover] is still kind of
basing it on what the skeptic is going to
0:13:44.750,0:13:45.800
pick next.
0:13:45.800,0:13:49.100
Vipul: Okay. It's actually pretty much the
same problem [as the
0:13:49.100,0:13:55.449
preceding one], in a somewhat milder form.
The prover is making
0:13:55.449,0:13:59.959
cases based on what the skeptic is going to
do next, and choosing a
0:13:59.959,0:14:01.940
strategy according to that. But the prover
doesn't actually know what
0:14:01.940,0:14:05.089
the skeptic is going to do next, so the prover
should actually have a
0:14:05.089,0:14:08.970
single strategy that works in both cases.
So cases will be made to
0:14:08.970,0:14:12.209
prove that the strategy works but the prover
has to have a single
0:14:12.209,0:14:12.459
strategy.
0:14:12.449,0:14:15.370
Now in this case the correct way of doing the proof is just, the
0:14:15.370,0:14:18.779
prover can pick delta as epsilon because that
will work in both cases.
0:14:18.779,0:14:20.019
KM: Exactly.
0:14:20.019,0:14:23.320
Vipul: Yeah. But in general if you have two
different piece
0:14:23.320,0:14:26.579
definitions then the way you would do it so
you would pick delta as
0:14:26.579,0:14:30.300
the min [minimum] of the deltas that work in
the two different pieces,
0:14:30.300,0:14:32.910
because you want to make sure that
both cases are covered. But
0:14:32.910,0:14:36.730
the point is you have to do that -- take
the min use that rather than
0:14:36.730,0:14:39.730
just say, "I'm going to choose my delta
based on what the skeptic is
0:14:39.730,0:14:42.589
going to move next." Okay?
0:14:42.589,0:14:49.120
So this is a milder form of the same
misconception that that was there in
0:14:49.120,0:14:56.120
the previous example we saw.
0:15:04.620,0:15:11.620
So, this is what I call the mildly telepathic
prover, right? The
0:15:14.970,0:15:18.579
prover is still behaving telepathically
predicting the skeptic's future
0:15:18.579,0:15:23.740
moves but it's not so bad. The prover is
just making, like, doing a
0:15:23.740,0:15:25.470
coin toss type of telepathy. Whereas in the
earlier one is prover is
0:15:25.470,0:15:30.790
actually, deciding exactly what x the skeptic
would pick. But it's still
0:15:30.790,0:15:32.790
the same problem and the reason why I think
people will have this
0:15:32.790,0:15:36.329
misconception is because they don't think
about it in terms of the
0:15:36.329,0:15:38.970
sequence in which the moves are made, and
the information that each
0:15:38.970,0:15:45.970
party has at any given stage of the game.
0:15:50.889,0:15:57.889
Let's do this one.
0:16:10.930,0:16:15.259
So, this is a limit claim, right? It says
that the limit as x approaches
0:16:15.259,0:16:22.259
1 of 2x is 2, okay? How do we go about showing
this? Well, the idea is
0:16:23.699,0:16:27.990
let's play the game, right? Let's say
the skeptic picks epsilon as
0:16:27.990,0:16:34.990
0.1, okay? The prover picks delta as 0.05.
The skeptic is when picking
0:16:35.139,0:16:38.790
epsilon as 0.1, the skeptic is saying, "Please
trap the function
0:16:38.790,0:16:43.800
between 1.9 and 2.1. Okay? Find the delta
small enough so that the
0:16:43.800,0:16:48.389
function value is trapped between 1.9 and
2.1. The prover picks delta
0:16:48.389,0:16:55.389
as 0.05 which means the prover is now getting
the input value trapped
0:16:57.850,0:17:04.850
between 0.95 and 1.05. That's 1 plus minus
this thing. And now the
0:17:05.439,0:17:09.070
prover is claiming that if the x value is
within this much distance of
0:17:09.070,0:17:13.959
1 except the value equal to 1, then the function
value is within 0.1
0:17:13.959,0:17:17.630
distance of 2. So, the skeptic tries picking
x within the interval
0:17:17.630,0:17:23.049
specified by the prover, so maybe the skeptic
picks 0.97 which is
0:17:23.049,0:17:26.380
within 0.05 distance of 1.
0:17:26.380,0:17:31.570
And then they check that 2x [the function f(x)] is
1.94, that is at the distance of 0.06
0:17:31.570,0:17:38.570
from 2. So, it's within 0.1 of the claimed
limit 2. So who won the game?
0:17:38.780,0:17:42.650
If the thing is within the interval then who
wins?
0:17:42.650,0:17:43.320
KM: The prover.
0:17:43.320,0:17:46.720
Vipul: The prover wins, right? So, the prover
won the game so therefore
0:17:46.720,0:17:52.100
this limit statement is true, right? So, what's
wrong with this as a
0:17:52.100,0:17:57.370
proof that the limit statement is true? How
is this not a proof that
0:17:57.370,0:18:03.870
the limit statement is true? This what I've
written here, why is that
0:18:03.870,0:18:05.990
not a proof that the limit statement is true?
0:18:05.990,0:18:11.960
KM: Because it's only an example for the
specific choice of epsilon and x.
0:18:11.960,0:18:16.200
Vipul: Yes, exactly. So, it's like a single
play of the game, the
0:18:16.200,0:18:20.470
prover wins, but the limit statement doesn't
just say that the prover
0:18:20.470,0:18:24.380
wins the game, it says the prover has a winning
strategy. It says that
0:18:24.380,0:18:27.660
the prover can win the game regardless of
how the skeptic plays;
0:18:27.660,0:18:31.070
there's a way for the prover to do that.
This just gives one example
0:18:31.070,0:18:34.640
where the prover won the game, but it doesn't
tell us that regardless
0:18:34.640,0:18:37.280
of the epsilon the skeptic picks the prover
can pick a delta such that
0:18:37.280,0:18:41.090
regardless of the x the skeptic picks, the
function is within the
0:18:41.090,0:18:45.530
thing. So that's the issue here. Okay?
0:18:45.530,0:18:51.160
Now you notice -- I'm sure you've noticed
this but the way the game and the
0:18:51.160,0:18:58.160
limit definition. The way the limit definition
goes, you see that all
0:18:59.870,0:19:04.260
the moves of the skeptic we write "for every"
"for all." Right? And
0:19:04.260,0:19:07.390
for all the moves of the prover we write "there
exists." Why do we do
0:19:07.390,0:19:11.140
that? Because we are trying to get a winning
strategy for the prover,
0:19:11.140,0:19:14.309
so the prover controls his own moves. Okay?
0:19:14.309,0:19:15.250
KM: Exactly.
0:19:15.250,0:19:18.630
Vipul: So, therefore wherever it's a prover
move it will be a there
0:19:18.630,0:19:22.240
exists. Where there is a skeptic's move
the prover has to be prepared
0:19:22.240,0:19:29.240
for anything the skeptic does. All those moves
are "for every."
0:19:30.559,0:19:33.850
One last one. By the way, this one was called,
"You say you want a
0:19:33.850,0:19:36.870
replay?" Which is basically they're just
saying that just one play is
0:19:36.870,0:19:40.890
not good enough. If the statement is actually
true, the prover should
0:19:40.890,0:19:45.370
be willing to accept it if the skeptic wants a
replay and say they want to
0:19:45.370,0:19:47.679
play it again, the prover should say "sure"
and "I'm going to win
0:19:47.679,0:19:53.320
again." That's what it would mean for
the limit statement to be true.
0:19:53.320,0:20:00.320
One last one. Just kind of pretty similar
to the one we just saw. But with
0:20:16.690,0:20:23.690
a little twist.
0:20:39.020,0:20:46.020
Okay, this one, let's see. We are saying
that the limit as x
0:20:50.450,0:20:56.900
approaches zero of sin(1/x) is zero, right?
Let's see how we prove
0:20:56.900,0:21:01.409
this. If the statement true ... well, do you
think the statement is
0:21:01.409,0:21:08.409
true? As x approach to zero, is sin 1 over
x approaching zero? So
0:21:13.980,0:21:20.980
here's the picture of sin(1/x). y-axis.
It's an oscillatory function
0:21:22.010,0:21:27.870
and it has this kind of picture. Does it doesn't
go to zero as x
0:21:27.870,0:21:29.270
approaches zero?
0:21:29.270,0:21:30.669
KM: No.
0:21:30.669,0:21:35.539
Vipul: No. So, you said that this statement
is false, but I'm going to
0:21:35.539,0:21:38.700
try to show it's true. Here's how I do
that. Let's say the skeptic
0:21:38.700,0:21:44.510
picks epsilon as two, okay? And then the prover
... so, the epsilon is
0:21:44.510,0:21:48.520
two so that's the interval of width two
about the game limit zero. The
0:21:48.520,0:21:55.150
prover picks delta as 1/pi. Whatever x the
skeptic picks, okay?
0:21:55.150,0:22:02.150
Regardless of the x that the
skeptic picks, the function is trapped
within epsilon of the game limit. Is that
0:22:10.340,0:22:16.900
true? Yes, because sin
(1/x) is between minus 1 and 1, right? Therefore
0:22:16.900,0:22:20.100
since the skeptic
picked an epsilon of 2, the function value
0:22:20.100,0:22:24.030
is completely trapped in
the interval from -1 to 1, so therefore the
0:22:24.030,0:22:27.919
prover managed to trap it
within distance of 2 of the claimed limit zero.
0:22:27.919,0:22:30.970
Okay? Regardless of what
the skeptic does, right? It's not just saying
0:22:30.970,0:22:34.370
that the prover won the
game once, it's saying whatever x the skeptic
0:22:34.370,0:22:40.740
picks the prover can
still win the game. Right? Regardless if the
0:22:40.740,0:22:43.780
x the skeptic picks, the
prover picked a delta such that the function
0:22:43.780,0:22:48.100
is trapped. It's
completely trapped, okay? It's not an issue
0:22:48.100,0:22:51.130
of whether the skeptic
picked a stupid x. Do you think that this
0:22:51.130,0:22:52.130
proves the statement?
0:22:52.130,0:22:59.130
KM: No, I mean in this case it still depended
on the epsilon that the
0:23:01.030,0:23:01.820
skeptic chose.
0:23:01.820,0:23:04.980
Vipul: It's still dependent on the epsilon
that the skeptic chose? So,
0:23:04.980,0:23:05.679
yes, that's exactly the problem.
0:23:05.679,0:23:09.370
So, we proved that the statement -- we prove
that from this part onward
0:23:09.370,0:23:12.500
but it still, we didn't prove it for all
epsilon, we only prove for
0:23:12.500,0:23:16.309
epsilon is 2, and 2 is a very big number,
right? Because the
0:23:16.309,0:23:19.970
oscillation is all happening between minus
1 and 1, and if in fact the
0:23:19.970,0:23:26.970
skeptic had pick epsilon as 1 or something
smaller than 1 then the two
0:23:27.030,0:23:32.169
epsilon strip width would not cover the entire
-1, +1
0:23:32.169,0:23:35.490
interval, and then whatever the prover did
the skeptic could actually
0:23:35.490,0:23:39.530
pick an x and show that it's not trapped.
So, in fact the reason why
0:23:39.530,0:23:43.110
the prover could win the game from this point
onward is that the
0:23:43.110,0:23:45.900
skeptic made a stupid choice of epsilon.
Okay?
0:23:45.900,0:23:52.289
In all these situation, all these misconceptions,
the main problem is,
0:23:52.289,0:23:58.919
that we're not ... keeping in mind the order
which the moves I made
0:23:58.919,0:24:04.179
and how much information each claim has at
the stage where that move
0:24:04.179,0:24:04.789
is being made.
Strongly telepathic prover
Spot the error in this:
Consider the limit problem . The proof corresponding to this problem would involve a game between a prover and a skeptic. To show that the limit statement is true, it suffices to exhibit a winning strategy for the prover for the game. The strategy is as follows. Pick . Let's prove that this works.
Specific claim: For any skeptic-picked , if the prover picks such that , then regardless of the that the skeptic picks with , we have .
Recall the sequence of moves made in the game. First, the skeptic picks . Then, the prover picks . Then, the skeptic picks a test value of to challenge the prover's claim of the function being trapped. In particular, this means that at the stage that the prover picks the value of , the prover is unaware of the value of that the skeptic plans to pick. Hence, the prover's strategy for choosing cannot use . Rather, the prover must have a strategy for purely in terms of , which is the only piece of information known to the prover at that stage in the game.
This also explains why we called this error the strongly telepathic prover, i.e., it involved the prover reading the skeptic's mind about future planned moves, which is impermissible.
Although this strategy is wrong, it can be fixed to get a correct strategy, i.e., this is the right way to start thinking about how this type of problem could be attacked. What the prover needs to do is pick a choice of that works for all that the skeptic can pick in the constrained interval. The algebra done here provides some guidelines on how the prover can make such a choice, but another idea, namely, the idea of a cut-off value, is needed to complete the strategy.
Mildly telepathic prover
Spot the error in this:
Consider the limit problem:
We want to show that For this game, we need to exhibit a winning strategy for the prover. The winning strategy is as follows. The skeptic first chooses . The prover now makes two cases. If the skeptic is planning to pick a rational value of , then the prover chooses the strategy . If the skeptic is planning to choose an irrational value of , then the prover can pick any . Clearly, the prover's strategy works in both cases, so we have a winning strategy.
Recall the sequence of moves made in the game. First, the skeptic picks . Then, the prover picks . Then, the skeptic picks a test value of to challenge the prover's claim of the function being trapped. In particular, this means that at the stage that the prover picks the value of , the prover is unaware of the value of that the skeptic plans to pick. Hence, the prover's strategy for choosing cannot rely on specifics about what the skeptic plans to choose.
This error is similar to the preceding error. Both involve impermissible telepathy on the prover's part in reading the skeptic's mind. The strongly telepathic prover error is more severe in the sense that it involves the prover reading the exact value of that the skeptic plans to play, whereas the mildly telepathic prover error only involves the prover guessing the type of value (rational or irrational) that the skeptic plans to play.
The fix for the mildly telepathic prover error is that the prover chooses a combined strategy that simultaneously works for both eventualities. In this situation, the strategy works for both situations (rational and irrational ). In general, for a function with two piece definitions for rational and irrational points in the domain, we need to take the min of the -strategies that work for the definitions individually. A similar approach works for different definitions on the left and right.
You say you want a replay?
Spot the error in this:
Consider the limit problem . Let's think of this in terms of an game. The skeptic begins by picking . The prover chooses . The skeptic now chooses . This value of is within the -distance of . It's now checked that is within -distance of the claimed limit . The prover has thus won the game, and we have established the truth of the limit statement.
This involves only one play of the limit game. The prover did win this play of the game. However, for us to declare the limit statement to be true, we need to establish that the prover has a winning strategy for the game, which means we need to demonstrate how the prover would pick a in terms of each choice of (preferably by specifying explicitly as a function of ) and then show that the strategy works for all within -distance of the point on the domain side.
It so happens that in this case, the limit statement is true and the prover did play the game according to one possible winning strategy: . However, since we weren't actually told the winning strategy, let alone given an explanation of why it works, what we're given fails as a proof.
Playing to lose
Spot the error in this:
Here's an easy proof that . We need to show that the prover has a winning strategy for the game. Let's say the skeptic starts out by picking . The prover then picks . It can now easily be verified that for , , because the function is trapped within . Thus, the prover has succeeded in trapping the function within the $\varepsilon$-interval specified by the skeptic, and hence won the game. The limit statement is therefore true.
This involves only one choice of . The proof does show that with the choice , the prover wins the game. However, in order to show that the limit statement is true, one would need to demonstrate that the prover wins the game for every possible choice of . In particular, from the skeptic's viewpoint, smaller is smarter, so the prover needs to have a strategy to win the game for arbitrarily small .
In fact, the limit statement is false, and for any choice of , the prover cannot win the game, because the range of the function on the immediate left and immediate right of zero is .
Conceptual definition and various cases
Formulation of conceptual definition
Below is the conceptual definition of limit. Suppose is a function defined in a neighborhood of the point , except possibly at the point itself. We say that:
if:
For every choice of neighborhood of (where the term neighborhood is suitably defined)
there exists a choice of neighborhood of (where the term neighborhood is suitably defined) such that
0:00:15.570,0:00:19.570
Vipul: Ok, so in this talk I'm going to
do the conceptual definition
0:00:19.570,0:00:26.320
of limit, which is important for a number
of reasons. The main reason
0:00:26.320,0:00:31.349
is it allows you to construct definitions
of limit, not just for this
0:00:31.349,0:00:34.430
one variable, function of one variable, two
sided limit which you have
0:00:34.430,0:00:38.930
hopefully seen before you saw this video.
Also for a number of other
0:00:38.930,0:00:43.210
limit cases which will include limits to infinity,
functions of two
0:00:43.210,0:00:47.789
variables, etc. So this is a general blueprint
for thinking about
0:00:47.789,0:00:54.789
limits. So let me put this definition here
in front for this. As I am
0:00:54.890,0:00:59.289
going, I will write things in more general.
So the starting thing is...
0:00:59.289,0:01:03.899
first of all f should be defined around the
point c, need not be
0:01:03.899,0:01:08.810
defined at c, but should be defined everywhere
around c. I won't write
0:01:08.810,0:01:11.750
that down, I don't want to complicate things
too much. So we start
0:01:11.750,0:01:18.750
with saying for every epsilon greater than
zero. Why are we picking
0:01:19.920,0:01:21.689
this epsilon greater than zero?
0:01:21.689,0:01:22.790
Rui: Why?
0:01:22.790,0:01:26.070
Vipul: What is the goal of this epsilon? Where
will it finally appear?
0:01:26.070,0:01:28.520
It will finally appear here. Is this captured?
0:01:28.520,0:01:29.520
Rui: Yes.
0:01:29.520,0:01:32.920
Vipul: Which means what we actually are picking
when we...if you've
0:01:32.920,0:01:37.720
seen the limit as a game video or you know
how to make a limit as a
0:01:37.720,0:01:41.700
game. This first thing has been chosen by
the skeptic, right, and the
0:01:41.700,0:01:45.840
skeptic is trying to challenge the prover
into trapping f(x) within L - epsilon to
0:01:45.840,0:01:50.210
L + epsilon. Even if you haven't
seen that [the game], the main focus of
0:01:50.210,0:01:55.570
picking epsilon is to pick this interval surrounding
L. So instead of
0:01:55.570,0:02:02.570
saying, for every epsilon greater than zero,
let's say for every
0:02:04.259,0:02:11.259
choice of neighborhood of L. So what I mean
by that, I have not
0:02:19.650,0:02:23.760
clearly defined it so this is a definition
which is not really a
0:02:23.760,0:02:28.139
definition, sort of the blueprint for definitions.
It is what you fill
0:02:28.139,0:02:31.570
in the details [of] and get a correct definition.
So by neighborhood,
0:02:31.570,0:02:36.180
I mean, in this case, I would mean something
like (L - epsilon, L +
0:02:36.180,0:02:43.180
epsilon). It is an open interval surrounding
L. Ok, this one. The
0:02:44.590,0:02:47.160
conceptual definition starts for every choice
of neighborhood of
0:02:47.160,0:02:54.160
L. The domain neighborhood, I haven't really
defined, but that is the
0:02:58.359,0:03:05.359
point, it is the general conceptual definition.
There exists...what
0:03:09.810,0:03:11.530
should come next? [ANSWER!]
0:03:11.530,0:03:16.530
Rui: A delta?
Vipul: That is what the concrete definition
0:03:16.530,0:03:18.530
says, but what would the
conceptual thing say?
0:03:18.530,0:03:21.680
Rui: A neighborhood.
Vipul: Of what? [ANSWER!]
0:03:21.680,0:03:28.680
Rui: Of c.
Vipul: Of c, of the domain. The goal of picking
0:03:34.639,0:03:37.970
delta is to find a
neighborhood of c. Points to the immediate
0:03:37.970,0:03:44.919
left and immediate
right of c. There exists a choice of neighborhood
0:03:44.919,0:03:51.919
of c such that, by
the way I sometimes abbreviate, such that,
0:03:59.850,0:04:06.109
as s.t., okay, don't get
confused by that. Okay, what next? Let's
0:04:06.109,0:04:12.309
bring out the thing. The next
thing is for all x with |x - c| less than
0:04:12.309,0:04:19.309
... all x in the neighborhood
except the point c itself. So what should
0:04:20.040,0:04:27.040
come here? For all x in the
neighborhood of c, I put x not equal to c.
0:04:36.570,0:04:37.160
Is that clear?
0:04:37.160,0:04:37.520
Rui: Yes.
0:04:37.520,0:04:44.520
Vipul: x not equal to c in the neighborhood
chosen for c. The reason
0:04:49.310,0:04:53.360
we're excluding the point c that we take the
limit at the point and we
0:04:53.360,0:04:55.770
just care about stuff around, we don't care
about what is happening at
0:04:55.770,0:05:02.770
the point. For c...this chosen neighborhood...I
am writing the black
0:05:09.880,0:05:14.440
for choices that the skeptic makes and the
red for the choices the
0:05:14.440,0:05:16.490
prover makes, actually that's reverse of what
I did in the other
0:05:16.490,0:05:21.320
video, but that's ok. They can change colors.
If you have seen that
0:05:21.320,0:05:24.710
limit game thing, this color pattern just
[means] ... the black
0:05:24.710,0:05:28.400
matches with the skeptic choices and the red
matches what the prover
0:05:28.400,0:05:32.710
chooses. If you haven't seen that, it is
not an issue. Just imagine
0:05:32.710,0:05:35.820
it's a single color.
0:05:35.820,0:05:40.820
What happens next? What do we need to check
in order to say this limit
0:05:40.820,0:05:42.950
is L? So f(x) should be where?
0:05:42.950,0:05:44.980
Rui: In the neighborhood of L.
0:05:44.980,0:05:48.060
Vipul: Yeah. In the concrete definition we
said f(x) minus L is less
0:05:48.060,0:05:51.440
than epsilon. Right, but that is just stating
that f(x) is in the
0:05:51.440,0:05:58.440
chosen neighborhood. So f(x) is in the chosen
neighborhood of...Now
0:06:08.470,0:06:15.470
that we have this blueprint for the definition.
This is a blueprint
0:06:25.660,0:06:32.660
for the definition. We'll write it in blue.
What I mean is, now if I
0:06:34.930,0:06:40.700
ask you to define a limit, in a slightly different
context; you just
0:06:40.700,0:06:46.280
have to figure out in order to make this rigorous
definition. What
0:06:46.280,0:06:49.240
word do you need to understand the meaning
of? [ANSWER!]
0:06:49.240,0:06:53.780
Rui: Neighborhood.
Vipul: Neighborhood, right. That's the magic
0:06:53.780,0:06:59.810
word behind which I am
hiding the details. If you can understand
0:06:59.810,0:07:06.280
what I mean by neighborhood
then you can turn this into a concrete definition.
Functions of one variable case
The following definitions of neighborhood are good enough to define limits.
For points in the interior of the domain, for functions of one variable: We can take an open interval centered at the point. For a point , such an open interval is of the form . Note that if we exclude the point itself, we get .
For the point , for functions of one variable: We take intervals of the form , where .
For the point , for functions of one variable: We can take interval of the form , where .
We can now list the nine cases of limits, combining finite and infinite possibilities:
Case
Definition
For every , there exists such that for all satisfying (i.e., ), we have (i.e., ).
For every , there exists such that for all satisfying (i.e., ), we have (i.e., ).
For every , there exists such that for all satisfying (i.e., ), we have (i.e., ).
For every , there exists such that for all satisfying (i.e., ), we have (i.e., ).
For every , there exists such that for all satisfying (i.e., ), we have (i.e., ).
For every , there exists such that for all satisfying (i.e., ), we have (i.e., ).
For every , there exists such that for all satisfying (i.e., ), we have (i.e., ).
For every , there exists such that for all satisfying (i.e., ), we have (i.e., ).
For every , there exists such that for all satisfying (i.e., ), we have (i.e., ).
Limit of sequence versus real-sense limit
Recall that the limit of a real-valued function to infinity is defined as follows:
means that:
For every
there exists (we're thinking of the neighborhood ) such that
for all (i.e. )
we have (i.e., ).
Suppose now instead that is a function restricted to the natural numbers. We can think of as a sequence, namely the sequence . In that case:
(in words, the sequence converges to ) means that:
For every
there exists such that
for all satisfying ,
we have (i.e., ).
The definitions differ both in their second and third line. However, the difference in the second line (the use of a real number versus a natural number to specify the threshold for the trapping interval) is not important, i.e., we could swap these lines between the definitions without changing the sense of either definition. The key difference between the definitions lies in their third lines. In the real-sense limit definition case, we require trapping of the function value close to the claimed limit for all sufficiently large reals whereas the sequence limit definition requires trapping only for all sufficiently large natural numbers.
To understand this distinction, consider the following: if is defined on reals, and it has a real-sense limit, i.e., for some , then it must also be true that . However, it is possible for to have a sequence limit but not have a real-sense limit. For instance, the function has undefined but is zero, because takes the value 0 at all integers.
0:00:15.530,0:00:22.530
Vipul: Okay. So this talk is going to be about
limit at infinity for functions on real numbers
0:00:24.300,0:00:28.980
and the concept of limits of sequences, how
these definitions are essentially almost the
0:00:28.980,0:00:34.790
same thing and how they differ.
0:00:34.790,0:00:41.790
Okay. So let's begin by reviewing the definition
of the limit as x approaches infinity of f(x).
0:00:42.360,0:00:47.390
Or rather what it means for that limit to
be a number L. Well, what it means is that
0:00:47.390,0:00:52.699
for every epsilon greater than zero, so we
first say for every neighborhood of L, small
0:00:52.699,0:00:59.429
neighborhood of L, given by radius epsilon
there exists a neighborhood of infinity which
0:00:59.429,0:01:03.010
is specified by choosing some a such that
that is
0:01:03.010,0:01:08.670
the interval (a,infinity) ...
0:01:08.670,0:01:15.220
... such that for all x in the interval from
a to infinity. That is for all x within the
0:01:15.220,0:01:20.430
chosen neighborhood of infinity, the f(x)
value is within the chosen neighborhood of
0:01:20.430,0:01:23.390
L. Okay?
0:01:23.390,0:01:28.049
If you want to think about it in terms of
the game between the prover and the skeptic,
0:01:28.049,0:01:34.560
the prover is claiming that the limit as x
approaches infinity of f(x) is L. The skeptic
0:01:34.560,0:01:38.930
begins by picking a neighborhood of L which
is parameterized by its radius epsilon. The
0:01:38.930,0:01:41.619
prover picks the
neighborhood of infinity which is parameterized
0:01:41.619,0:01:48.350
by its lower end a. Then the skeptic picks
a value x between a and infinity. Then they
0:01:48.350,0:01:51.990
check whether absolute value f(x) minus L
[symbolically: |f(x) - L|] is less than epsilon.
0:01:51.990,0:01:56.090
That is they check whether f(x) is in the
chosen neighborhood of L (the neighborhood
0:01:56.090,0:02:00.640
chosen by the skeptic). If it is,
then the prover wins. The prover has managed
0:02:00.640,0:02:05.810
to trap the function: for x large enough,
the prover has managed to trap the function
0:02:05.810,0:02:12.810
within epsilon distance of L. If not, then
the skeptic wins. The statement is true if
0:02:13.610,0:02:18.680
the prover has a winning the strategy for
the game.
0:02:18.680,0:02:21.730
Now, there is a similar definition which one
has for sequences. So, what's a sequence?
0:02:21.730,0:02:26.349
Well, it's just a function from the natural
numbers. And, here, we're talking of sequences
0:02:26.349,0:02:31.610
of real numbers. So, it's a function from
the naturals to the reals and we use the same
0:02:31.610,0:02:37.400
letter f for a good reason. Usually we write
sequences with subscripts, a_n type of thing.
0:02:37.400,0:02:42.409
But I'm using it as a function just to highlight
the similarities. So, limit as n approaches
0:02:42.409,0:02:47.519
infinity, n restricted to the natural numbers
... Usually if it's clear we're talking of
0:02:47.519,0:02:52.830
a sequence, we can remove this part [pointing
to the n in N constraint specification] just
0:02:52.830,0:02:54.980
say limit n approaches infinity f(n),
but since we want to be really clear here,
0:02:54.980,0:02:57.220
I have put this line. Okay?
0:02:57.220,0:03:02.709
So, this limit equals L means "for every epsilon
greater than 0 ..." So, it starts in the same
0:03:02.709,0:03:09.170
way. The skeptic picks a neighborhood of L.
Then the next line is a little different but
0:03:09.170,0:03:16.170
that's not really the crucial part. The skeptic
is choosing epsilon. The prover picks n_0,
0:03:18.799,0:03:22.830
a natural number. Now, here the prover is
picking a real number. Here the prover is
0:03:22.830,0:03:26.700
picking a natural number. That's not really
the big issue. You could in fact change this
0:03:26.700,0:03:33.659
line to match. You could interchange these
lines. It wouldn't affect either definition.
0:03:33.659,0:03:40.599
The next line is the really important one
which is different. In here [pointing to real-sense
0:03:40.599,0:03:47.430
limit], the condition has to be valid for
all x, for all real numbers x which are bigger
0:03:47.430,0:03:51.900
than the threshold which the prover has chosen.
Here on the other hand [pointing to the sequence
0:03:51.900,0:03:56.970
limit] the condition has to be valid for all
natural numbers which are bigger than the
0:03:56.970,0:04:00.659
threshold the prover has chosen. By the way,
some of you may have seen the definition with
0:04:00.659,0:04:07.659
an equality sign here. It doesn't make a difference
to the definition. It does affect what n_0
0:04:09.010,0:04:12.019
you can choose, it will go up or down by one,
but that's not
0:04:12.019,0:04:17.310
really a big issue. The big issue, the big
difference between these two definitions is
0:04:17.310,0:04:23.050
that in this definition you are insisting
that the condition here is valid for all real
0:04:23.050,0:04:30.050
x. So, you are insisting or rather the game
is forcing the prover to figure out how to
0:04:31.650,0:04:36.940
trap the function values for all real x. Whereas
here, the game is only requiring the prover
0:04:36.940,0:04:39.639
to trap the function values for all large
enough
0:04:39.639,0:04:42.880
natural numbers. So, here [real-sense limit]
it's all large enough real numbers. Here [sequence
0:04:42.880,0:04:49.250
limit] it's all large enough natural numbers.
Okay?
0:04:49.250,0:04:56.250
So, that's the only difference essentially.
Now, you can see from the way we have written
0:04:57.050,0:04:59.900
this that this [real-sense limit] is much
stronger. So, if you do have a function which
0:04:59.900,0:05:06.880
is defined on real so that both of these concepts
can be discussed. If it were just a sequence
0:05:06.880,0:05:10.080
and there were no function to talk about then
obviously, we can't even talk about this.
0:05:10.080,0:05:16.860
If there's a function defined on the reals
or on all large enough reals, then we can
0:05:16.860,0:05:21.470
try taking both of these. The existence of
this [pointing at the real-sense limit] and
0:05:21.470,0:05:24.580
[said "or", meant "and"] it's being equal
to L as much stronger than this [the sequence
0:05:24.580,0:05:27.250
limit] equal to L. If this is equal to L then
definitely this [the sequence limit] is equal
0:05:27.250,0:05:29.330
to L. Okay?
0:05:29.330,0:05:32.080
But maybe there are situations where this
[the sequence limit] is equal to some number
0:05:32.080,0:05:38.240
but this thing [the real-sense limit] doesn't
exist. So, I want to take one example here.
0:05:38.240,0:05:45.240
I have written down an example and we can
talk a bit about that is this. So, here is
0:05:45.509,0:05:52.509
a function. f(x) = sin(pi x). This is sin
(pi x) and the corresponding
0:05:55.630,0:06:00.530
function if you just restrict [it] to the
natural numbers is just sin (pi n). Now, what
0:06:00.530,0:06:06.759
does sin (pi n) look like for a natural number
n? In fact for any integer n? pi times
0:06:06.759,0:06:13.759
n is an integer multiple of pi. sin of integer
multiples of pi is zero. Let's make a picture
0:06:18.370,0:06:25.370
of sin ...
0:06:27.289,0:06:33.360
It's oscillating. Right? Integer multiples
of pi are precisely the ones where it's meeting
0:06:33.360,0:06:40.330
the axis. So, in fact we are concerned about
the positive one because we are talking of
0:06:40.330,0:06:45.840
the sequence (natural number [inputs]). Okay?
And so, if you are looking at this sequence,
0:06:45.840,0:06:51.090
all the terms here are zero. So, the limit
is also zero. So, this limit [the sequence
0:06:51.090,0:06:53.030
limit] is zero.
0:06:53.030,0:07:00.030
Okay. What about this limit? Well, we have
the picture again. Is it going anywhere? No.
0:07:05.349,0:07:07.650
It's oscillating between minus one and one
[symbolically: oscillating in [-1,1]]. It's
0:07:07.650,0:07:11.669
not settling down to any number. It's not...
You cannot trap it near any particular number
0:07:11.669,0:07:17.280
because it's all over the map between minus
one and one. For the same reason that sin(1/x)
0:07:17.280,0:07:22.840
doesn't approach anything as x approaches
zero, the same reason sin x or sin(pi x) doesn't
0:07:22.840,0:07:29.840
approach anything as x approaches infinity.
So, the limit for the real thing, this does
0:07:31.099,0:07:37.539
not exist. So, this gives an example where
the real thing [the real-sense limit] doesn't
0:07:37.539,0:07:44.539
exist and the sequence thing [sequence limit]
does exist and so here is the overall summary.
0:07:44.690,0:07:46.979
If the real sense limit,
that is this one [pointing to definition of
0:07:46.979,0:07:51.039
real sense limit] exists, [then] the sequence
limit also exists and they're both equal.
0:07:51.039,0:07:54.419
On the other hand, you can have a situation
with the real sense limit, the limit for the
0:07:54.419,0:08:00.819
function of reals doesn't exist but the sequence
limit still exists like this set up. Right?
0:08:00.819,0:08:05.569
Now, there is a little caveat that I want
to add. If the real sense limit doesn't exist
0:08:05.569,0:08:11.069
as a finite number but it's say plus infinity
then the sequence limit also has to be plus
0:08:11.069,0:08:16.150
infinity. If the real sense limit is minus
infinity, then the sequence limit also has
0:08:16.150,0:08:20.330
to be minus infinity. So, this type of situation,
where the real sense limit doesn't exist but
0:08:20.330,0:08:26.840
the sequence exists, well, will happen in
kind of oscillatory type of situations. Where
0:08:26.840,0:08:31.409
the real sense you have an oscillating thing
and in the sequence thing on the other hand
0:08:31.409,0:08:36.330
you somehow manage to pick a bunch of points
where that oscillation doesn't create a problem.
0:08:36.330,0:08:36.789
Okay?
0:08:36.789,0:08:43.630
Now, why is this important? Well, it's important
because in a lot of cases when you have to
0:08:43.630,0:08:50.630
calculate limits of sequences, you just calculate
them by doing, essentially, just calculating
0:08:53.230,0:09:00.230
the limits of the function defining the sequence
as a limit of a real valued function. Okay?
0:09:00.230,0:09:03.460
So, for instance if I ask you what is limit
...
0:09:03.460,0:09:10.460
Okay. I'll ask you what is limit [as] n approaches
infinity of n^2(n + 1)/(n^3 + 1) or something
0:09:15.200,0:09:22.200
like that. Right? Some rational function.
You just do this calculation as if you were
0:09:25.430,0:09:29.720
just doing a limit of a real function, function
of real numbers, right? The answer you get
0:09:29.720,0:09:33.060
will be the correct one. If it's a finite
number it will be the same finite number.
0:09:33.060,0:09:37.850
In this case it will just be one. But any
rational function, if the answer is finite,
0:09:37.850,0:09:44.070
same answer for the sequence. If it is plus
infinity, same answer for the sequence. If
0:09:44.070,0:09:46.250
it is minus infinity, same answer as for the
sequence.
0:09:46.250,0:09:53.250
However, if the answer you get for the real-sense
limit is oscillatory type of non existence,
0:09:54.660,0:09:59.410
then that's inconclusive as far as the sequence
is concerned. You actually have to think about
0:09:59.410,0:10:05.520
the sequence case and figure out for yourself
what happens to the limit. Okay? If might
0:10:05.520,0:10:07.230
in
fact be the case that the sequence limit actually
0:10:07.230,0:10:11.380
does exist even though the real sense [limit]
is oscillatory. Okay.
Real-valued functions of multiple variables case
We consider the multiple input variables as a vector input variable, as the definition is easier to frame from this perspective.
The correct notion of neighborhood is as follows: for a point , we define the neighborhood parametrized by a positive real number as the open ball of radius centered at , i.e., the set of all points such that the distance from to is less than . This distance is the same as the norm of the difference vector . The norm is sometimes denoted . This open ball is sometimes denoted .
Suppose is a real-valued (i.e., scalar) function of a vector variable . Suppose is a point such that is defined "around" , except possibly at . In other words, there is an open ball centered at such that is defined everywhere on that open ball, except possibly at .
With these preliminaries out of the way, we can define the notion of limit. We say that:
if the following holds:
For every
there exists such that
for all satisfying (i.e., is in a ball of radius centered at but not the point itself -- note that the notation is for the norm, or length, of a vector)