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Full timed transcript: <toggledisplay>0:00:15.549,0:00:19.259
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. That was 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
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closer and closer to c, f(x) is sort of hanging
around L. It’s coming
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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 its limit in
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this approach sense, which is the mathematical
meaning of limit.
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There is another sense in which the word limit
is used in the English
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language, which is limit as a boundary or
a as a gap or as a bound.
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We may say, there is a limit to how many apples
you can eat from the
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food vault or something, and that sense of
limit is not used ... for
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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
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smaller and smaller. This doesn’t quite
work unless your function is
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increasing or decreasing near c, so you could
have various
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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
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elaboration and without a clear definition.
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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,
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which you may have seen in school. (well,
if you’ve seen limits in
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school, which hopefully you have. This video
is sort of more of a
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review type than learning it for the first
time). Let's try to
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understand this from that point of view.
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Let's say, you have a function whose graph
looks something like this.
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This is x of c, so this is the value x of
c, and this is a graph of
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the function, these curves are the graph of
the function, so where x
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is less than c, the graph is along this curve.
For x greater than c,
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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 prospective Y coordinates, so this is
x, this is Y, this is the
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graph. Y is f(x). When x is to the initial
left of c, the value, Y
0:04:35.749,0:04:42.749
value, the Y approach 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
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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
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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
sides of 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
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limit is here; the right-hand limit is up
here.
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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 they're
headed to, and add 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
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values are 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 limits still exist 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, they're 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
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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
where 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 define 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 pure 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 secant 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 [inaudible 00:10:36] 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 here.
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 of 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 up
0:11:36.879,0:11:42.810
with an S [inaudible 00:11:21] 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 here 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 the other limit
is zero? Why? Well, it's
0:14:49.329,0:14:52.949
kind of balance around here. 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 that’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 with the minus 1 and 1. However,
smaller 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 this time 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 [inaudible 00:20:30].
0:20:46.660,0:20:52.060
That’s how it is coming, actually, 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
tracked 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 Epsilon 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.</toggledisplay>


==Definition for finite limit for function of one variable==
==Definition for finite limit for function of one variable==
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===Left hand limit===
===Left hand limit===

Revision as of 20:01, 22 December 2012

ORIGINAL FULL PAGE: Limit
STUDY THE TOPIC AT MULTIPLE LEVELS:
ALSO CHECK OUT: Quiz (multiple choice questions to test your understanding) |Page with videos on the topic, both embedded and linked to

Motivation and general idea

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Definition for finite limit for function of one variable

Two-sided limit

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Left hand limit

Right hand limit

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Relation between the limit notions

Definition of finite limit for function of one variable in terms of a game

Two-sided limit

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Non-existence of limit

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Misconceptions

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Conceptual definition and various cases

Formulation of conceptual definition

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Functions of one variable case

This covers limits at and to infinity.

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Real-valued functions of multiple variables case

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