Video:Limit: Difference between revisions

From Calculus
 
(31 intermediate revisions by the same user not shown)
Line 1: Line 1:
{{perspectives}}
{{perspectives}}


The videos below are all taken from certain playlists. Instead of watching the videos on this page, you may prefer to watch the entire playlists on YouTube. Below are the playlist links:
* [https://www.youtube.com/playlist?list=PL8483BCA409563C88&feature=view_all Limit: first time college pack] (7 videos)
* [https://www.youtube.com/playlist?list=PLC0bHnWu122lmsGOHv39OSaNwD8MXvlTH&feature=view_all Limit: conceptual and infinity pack] (3 videos)
* Other playlist links to be added
==Motivation and general idea==
==Motivation and general idea==


Line 14: Line 19:


0:00:24.619,0:00:28.099
0:00:24.619,0:00:28.099
epsilon-delta definition. That was just an intuitive idea,
epsilon-delta definition. This is just an intuitive idea,
and a few somewhat
and a few somewhat


Line 49: Line 54:
0:01:02.050,0:01:06.640
0:01:02.050,0:01:06.640
number c, f(x) approaches some number L, and
number c, f(x) approaches some number L, and
that’s what this is:
that's what this is:


0:01:06.640,0:01:09.030
0:01:06.640,0:01:09.030
Line 60: Line 65:
0:01:15.259,0:01:22.259
0:01:15.259,0:01:22.259
closer and closer to c, f(x) is sort of hanging
closer and closer to c, f(x) is sort of hanging
around L. It’s coming
around L. It's coming


0:01:22.410,0:01:28.720
0:01:22.410,0:01:28.720
Line 68: Line 73:
0:01:28.720,0:01:32.429
0:01:28.720,0:01:32.429
word limit is used in the English language:
word limit is used in the English language:
One meaning its limit in
One meaning is limit in


0:01:32.429,0:01:36.310
0:01:32.429,0:01:36.310
Line 79: Line 84:


0:01:41.319,0:01:46.220
0:01:41.319,0:01:46.220
language, which is limit as a boundary or
language, which is limit as a boundary or as a cap or as a bound.
a as a gap or as a bound.


0:01:46.220,0:01:53.160
0:01:46.220,0:01:53.160
Line 87: Line 91:


0:01:53.160,0:01:58.640
0:01:53.160,0:01:58.640
food vault or something, and that sense of
fruit bowl or something, and that sense of
limit is not used ... for
limit is not used ... for


Line 103: Line 107:


0:02:11.800,0:02:18.800
0:02:11.800,0:02:18.800
so we don’t get confused in mathematics.
so we don't get confused in mathematics.
As I said, the idea is that
As I said, the idea is that


Line 119: Line 123:


0:02:32.740,0:02:37.980
0:02:32.740,0:02:37.980
smaller and smaller. This doesn’t quite
smaller and smaller. This doesn't quite
work unless your function is
work unless your function is


Line 131: Line 135:


0:02:46.750,0:02:52.170
0:02:46.750,0:02:52.170
doesn’t really … it's not very clear what
doesn't really ... it's not very clear what
we mean here without further
we mean here without further


Line 147: Line 151:
0:03:09.180,0:03:13.430
0:03:09.180,0:03:13.430
which you may have seen in school. (well,
which you may have seen in school. (well,
if you’ve seen limits in
if you've seen limits in


0:03:13.430,0:03:17.110
0:03:13.430,0:03:17.110
Line 165: Line 169:


0:03:35.990,0:03:42.990
0:03:35.990,0:03:42.990
This is x of c, so this is the value x of
This is x is c, so this is the value x is
c, and this is a graph of
c, and this is the graph of


0:03:44.069,0:03:48.310
0:03:44.069,0:03:48.310
Line 196: Line 200:


0:04:17.819,0:04:23.259
0:04:17.819,0:04:23.259
values … so the function, the graph of it,
values ... so the function, the graph of it,
the function values are
the function values are


0:04:23.259,0:04:27.449
0:04:23.259,0:04:27.449
their prospective Y coordinates, so this is
their respective y coordinates, so this is
x, this is Y, this is the
x, this is y, this is the


0:04:27.449,0:04:34.449
0:04:27.449,0:04:34.449
graph. Y is f(x). When x is to the initial
graph. y is f(x). When x is to the immediate
left of c, the value, Y
left of c, the value, y


0:04:35.749,0:04:42.749
0:04:35.749,0:04:42.749
value, the Y approach f(x) value is … are
value, the y equals f(x) value is ... are
these values, so this or
these values, so this or


0:04:44.610,0:04:51.610
0:04:44.610,0:04:51.610
this. As x approaches c from the left, the
this. As x approaches c from the left, the
Y values are approaching
y values are approaching


0:04:53.699,0:04:57.240
0:04:53.699,0:04:57.240
the Y coordinate of this open circle.
the y coordinate of this open circle.


0:04:57.240,0:05:04.240
0:04:57.240,0:05:04.240
Line 224: Line 228:
0:05:05.680,0:05:10.830
0:05:05.680,0:05:10.830
approaching c from the left, then the limit
approaching c from the left, then the limit
would be the Y coordinate
would be the y coordinate


0:05:10.830,0:05:16.279
0:05:10.830,0:05:16.279
Line 231: Line 235:


0:05:16.279,0:05:22.749
0:05:16.279,0:05:22.749
right, so approaches from here … the Y coordinate
right, so approaches from here ... the y coordinate
is approaching the Y
is approaching the y


0:05:22.749,0:05:29.749
0:05:22.749,0:05:29.749
Line 252: Line 256:


0:05:58.089,0:06:05.089
0:05:58.089,0:06:05.089
right, that’s plus of f(x),
right, that's plus of f(x),
is L2, and the value f of c is some third
is L2, and the value f of c is some third


0:06:08.059,0:06:15.059
0:06:08.059,0:06:15.059
number. We don’t know what
number. We don't know what
it is, but f of c, L1, L2, are in this case
it is, but f of c, L1, L2, are in this case


Line 268: Line 272:
0:06:25.900,0:06:28.259
0:06:25.900,0:06:28.259
concept of limit is usually a concept of two
concept of limit is usually a concept of two
sides of limit, which
sided limit, which


0:06:28.259,0:06:33.419
0:06:28.259,0:06:33.419
Line 283: Line 287:


0:06:39.860,0:06:43.279
0:06:39.860,0:06:43.279
doesn’t matter, so whether the value exists,
doesn't matter, so whether the value exists,
what it is, does not
what it is, does not


Line 303: Line 307:


0:07:03.499,0:07:07.749
0:07:03.499,0:07:07.749
sort of that. For the left-hand limit, you
sort of that: For the left-hand limit, you
basically sort of follow
basically sort of follow


Line 311: Line 315:


0:07:11.499,0:07:15.789
0:07:11.499,0:07:15.789
get the Y coordinate of that. For the right-hand
get the y coordinate of that. For the right-hand
limit, you follow
limit, you follow


0:07:15.789,0:07:21.129
0:07:15.789,0:07:21.129
the graph on the right and see where they're
the graph on the right and see where we're
headed to, and add the Y
headed to, and get the y


0:07:21.129,0:07:22.240
0:07:21.129,0:07:22.240
Line 334: Line 338:


0:07:52.610,0:07:55.889
0:07:52.610,0:07:55.889
values are different. You could also have
value is different. You could also have
a situation where the value
a situation where the value


0:07:55.889,0:08:00.460
0:07:55.889,0:08:00.460
doesn’t exist at all. The function isn't
doesn't exist at all. The function isn't
defined at the point, but
defined at the point, but


0:08:00.460,0:08:03.139
0:08:00.460,0:08:03.139
the limits still exist because the left-hand
the limit still exists because the left-hand
limit and right-hand
limit and right-hand


Line 349: Line 353:


0:08:04.719,0:08:09.979
0:08:04.719,0:08:09.979
Now, all these examples, they're sort of a
Now, all these examples, there's sort of a
crude way of putting this
crude way of putting this


Line 365: Line 369:


0:08:23.929,0:08:28.259
0:08:23.929,0:08:28.259
that’s headed to, and use another finger
that's headed to, and use another finger
to trace the curve on the
to trace the curve on the


0:08:28.259,0:08:33.640
0:08:28.259,0:08:33.640
immediate right and see where that’s headed
immediate right and see where that's headed
to, and if your two
to, and if your two


0:08:33.640,0:08:38.270
0:08:33.640,0:08:38.270
fingers can meet each other, then the place
fingers can meet each other, then the place
where they meet, the Y
where they meet, the y


0:08:38.270,0:08:41.870
0:08:38.270,0:08:41.870
Line 385: Line 389:


0:08:46.940,0:08:51.120
0:08:46.940,0:08:51.120
is here, and then the limit doesn’t exist
is here, and then the limit doesn't exist
because the left-hand limit
because the left-hand limit


Line 393: Line 397:
0:08:53.509,0:08:59.819
0:08:53.509,0:08:59.819
This, hopefully, you have seen in great detail
This, hopefully, you have seen in great detail
where you’ve done
when you've done


0:08:59.819,0:09:05.779
0:08:59.819,0:09:05.779
Line 401: Line 405:
0:09:05.779,0:09:11.850
0:09:05.779,0:09:11.850
this two-finger test is not really a good
this two-finger test is not really a good
definition of limit. What’s
definition of limit. What's


0:09:11.850,0:09:13.600
0:09:11.850,0:09:13.600
Line 416: Line 420:


0:09:25.220,0:09:29.440
0:09:25.220,0:09:29.440
hard, and it doesn’t really solve any problem.
hard, and it doesn't really solve any problem.
It's not really a
It's not really a


Line 440: Line 444:
0:09:50.040,0:09:56.990
0:09:50.040,0:09:56.990
things could give us trouble? Why do we need
things could give us trouble? Why do we need
to define our
to refine our


0:09:56.990,0:10:03.209
0:09:56.990,0:10:03.209
Line 454: Line 458:


0:10:18.220,0:10:21.899
0:10:18.220,0:10:21.899
have to develop a pure cut concept of limit
have to develop a clear cut concept of limit
to be able to answer this
to be able to answer this


Line 466: Line 470:


0:10:32.920,0:10:39.920
0:10:32.920,0:10:39.920
just equal secant x. It's not that. It's sine
just equal cosecant x. It's not that. It's sine
of 1 over x, and this
of 1 over x, and this


Line 474: Line 478:


0:10:50.220,0:10:52.660
0:10:50.220,0:10:52.660
that that’s not defined, isn't good enough
that that's not defined, isn't good enough
for us to say the limit
for us to say the limit


0:10:52.660,0:10:55.139
0:10:52.660,0:10:55.139
doesn't [inaudible 00:10:36] we actually have
doesn't exist; we actually have
to try to make a picture
to try to make a picture


0:10:55.139,0:10:57.660
0:10:55.139,0:10:57.660
of this and try to understand what the limit
of this and try to understand what the limit
is here.
is going to be.


0:10:57.660,0:11:04.660
0:10:57.660,0:11:04.660
Line 490: Line 494:


0:11:12.560,0:11:19.560
0:11:12.560,0:11:19.560
will sine 1 over x look? Let's start of where
will sine 1 over x look? Let's start off where
x is nearly infinity.
x is nearly infinity.


Line 503: Line 507:
0:11:30.660,0:11:36.879
0:11:30.660,0:11:36.879
therefore slightly positive. It's like here.
therefore slightly positive. It's like here.
It's going to start up
It's going to start off


0:11:36.879,0:11:42.810
0:11:36.879,0:11:42.810
with an S [inaudible 00:11:21] at zero. Then
with an asymptote, a horizontal asymptote, at zero.  
it's going to sort of go
Then it's going to sort of go


0:11:42.810,0:11:49.420
0:11:42.810,0:11:49.420
Line 533: Line 537:


0:12:16.990,0:12:21.160
0:12:16.990,0:12:21.160
1 over 3 pi, and so on. What’s going to
1 over 3 pi, and so on. What's going to
happen is that near zero it's
happen is that near zero it's


Line 573: Line 577:


0:13:38.850,0:13:45.060
0:13:38.850,0:13:45.060
I’m not being very accurate here, but just
I'm not being very accurate here, but just
the idea. The pen or
the idea. The pen or


Line 597: Line 601:


0:14:18.050,0:14:21.579
0:14:18.050,0:14:21.579
this, this … you're sort of getting close
this, this ... you're sort of getting close
to here but still not quite
to zero but still not quite


0:14:21.579,0:14:28.579
0:14:21.579,0:14:28.579
reaching it. It's … where are you headed?
reaching it. It's ... where are you headed?
It's kind of a little
It's kind of a little


0:14:31.610,0:14:36.879
0:14:31.610,0:14:36.879
unclear. Notice, it's not that just because
unclear. Notice, it's not that just because
we plug in zero doesn’t
we plug in zero doesn't


0:14:36.879,0:14:39.170
0:14:36.879,0:14:39.170
make sense, the limit doesn't... That’s
make sense, the limit doesn't... That's
not the issue. The issue is
not the issue. The issue is


0:14:39.170,0:14:43.249
0:14:39.170,0:14:43.249
that after you make the graph, it's unclear
that after you make the graph, it's unclear
what’s happening.
what's happening.


0:14:43.249,0:14:49.329
0:14:43.249,0:14:49.329
One kind of logic is that the other limit
One kind of logic is that, yeah, the limit
is zero? Why? Well, it's
is zero? Why? Well, it's


0:14:49.329,0:14:52.949
0:14:49.329,0:14:52.949
kind of balance around here. It's a bit above
kind of balanced around zero, right? It's a bit
and below, and it keeps
above and below, and it keeps


0:14:52.949,0:14:59.949
0:14:52.949,0:14:59.949
Line 637: Line 641:


0:15:12.459,0:15:17.449
0:15:12.459,0:15:17.449
If you think of limit as something that’s
If you think of limit as something it's
approaching, then as x
approaching, then as x


Line 649: Line 653:


0:15:36.550,0:15:41.920
0:15:36.550,0:15:41.920
zero, any small … this you make around zero,
zero, any small ... this you make around zero,
the graph is going to
the graph is going to


Line 660: Line 664:


0:15:47.269,0:15:50.300
0:15:47.269,0:15:50.300
oscillating with the minus 1 and 1. However,
oscillating within [-1,1]. However
smaller interval you
small an interval you


0:15:50.300,0:15:54.540
0:15:50.300,0:15:54.540
Line 696: Line 700:


0:16:28.639,0:16:33.089
0:16:28.639,0:16:33.089
close and stay close. So that’s actually
close and stay close. So that's actually
key idea number two we have
key idea number two we have


0:16:33.089,0:16:38.290
0:16:33.089,0:16:38.290
here the function … for the function to
here the function ... for the function to
have a limit at the point, the
have a limit at the point, the


Line 711: Line 715:


0:16:45.079,0:16:49.459
0:16:45.079,0:16:49.459
This is, therefore, it doesn’t have a limit
This is, therefore, it doesn't have a limit
at zero because the
at zero because the


Line 719: Line 723:


0:16:54.420,0:17:01.059
0:16:54.420,0:17:01.059
trap the function values. You cannot say that…
trap the function values. You cannot say that...
you cannot trap the
you cannot trap the


Line 750: Line 754:


0:17:30.330,0:17:33.890
0:17:30.330,0:17:33.890
need to remember is that the function doesn’t
need to remember is that the function doesn't
just need to come close
just need to come close


Line 789: Line 793:


0:18:30.550,0:18:37.550
0:18:30.550,0:18:37.550
of … what’s close enough? Is 2.1 close
of ... what's close enough? Is 2.1 close
enough? No, that’s too far.
enough? No, that's too far.


0:18:38.750,0:18:43.380
0:18:38.750,0:18:43.380
Line 796: Line 800:


0:18:43.380,0:18:47.420
0:18:43.380,0:18:47.420
Now, if you weren’t a mathematician, you
Now, if you weren't a mathematician, you
would probably say, "Yes,
would probably say, "Yes,


Line 844: Line 848:
0:19:59.990,0:20:05.940
0:19:59.990,0:20:05.940
this picture, and I change it to, let's say
this picture, and I change it to, let's say
… so I replace this
... so I replace this


0:20:05.940,0:20:11.410
0:20:05.940,0:20:11.410
Line 870: Line 874:


0:20:32.040,0:20:35.000
0:20:32.040,0:20:35.000
behavior sort of at this time that point or
behavior, sort of at that point or
farther away than that
farther away than that


0:20:35.000,0:20:42.000
0:20:35.000,0:20:42.000
point, then the behavior close to 2 doesn’t
point, then the behavior close to 2 doesn't
get affected. That’s the
get affected. That's the


0:20:42.820,0:20:46.660
0:20:42.820,0:20:46.660
other key idea here. Actually I did these
other key idea here. Actually I did these
in [inaudible 00:20:30].
in reverse order.


0:20:46.660,0:20:52.060
0:20:46.660,0:20:52.060
That’s how it is coming, actually, but I'll
That's how it was coming naturally, but I'll
just say it again.
just say it again.


Line 890: Line 894:


0:20:56.570,0:21:00.210
0:20:56.570,0:21:00.210
doesn’t depend on the behavior at any single
doesn't depend on the behavior at any single
specific other point. It
specific other point. It


Line 910: Line 914:


0:21:19.790,0:21:26.790
0:21:19.790,0:21:26.790
tracked near the point for the limit notion
trapped near the point for the limit notion
to be true. This type of
to be true. This type of


Line 926: Line 930:


0:21:40.590,0:21:47.590
0:21:40.590,0:21:47.590
strip. In that case, the limit doesn’t exist.
strip. In that case, the limit doesn't exist.
In subsequent videos,
In subsequent videos,


0:21:48.550,0:21:54.630
0:21:48.550,0:21:54.630
we'll see Epsilon definition, we'll do a bit
we'll see the epsilon delta definition, we'll do a bit
of formalism to that, and
of formalism to that, and


Line 2,236: Line 2,240:
Vipul: Ok, so this talk is going to be about
Vipul: Ok, so this talk is going to be about
why under certain circumstances limits don't exist
why under certain circumstances limits don't exist
0:00:38.710,0:00:39.800


0:00:39.800,0:00:46.800
0:00:39.800,0:00:46.800
We are going to take this example of a function
We are going to take this example of a function
which is defined like this: sin of one over x
which is defined like this: sin of one over x
0:00:47.270,0:00:47.699


0:00:47.699,0:00:51.360
0:00:47.699,0:00:51.360
Line 2,841: Line 2,841:
==Misconceptions==
==Misconceptions==


<center>{{#widget:YouTube|id=Kms_VHwgdZ8}}</center>
<center>{{#widget:YouTube|id=F0r_offAc5M}}</center>
 
Full timed transcript: <toggledisplay>
0:00:15.500,0:00:19.140
Vipul: Okay. This talk is going to be about
certain misconceptions
 
0:00:19.140,0:00:22.440
that people have regarding limits and these
are misconceptions that
 
0:00:22.440,0:00:25.840
people generally acquire after...
 
0:00:25.840,0:00:29.180
These are not the misconceptions that
people have before studying limits,
 
0:00:29.180,0:00:32.730
these are misconceptions you might have after
studying limits,
 
0:00:32.730,0:00:35.059
after studying the epsilon delta definition.
 
0:00:35.059,0:00:38.550
I'm going to describe these misconceptions
in terms of the limit game,
 
0:00:38.550,0:00:41.900
the prover skeptic game of the limit. Though
the misconceptions
 
0:00:41.900,0:00:45.850
themselves don't depend on
the understanding of the
 
0:00:45.850,0:00:49.059
game but to understand exactly what's
happening, it's better to think
 
0:00:49.059,0:00:51.010
of it in terms of the game.
 
0:00:51.010,0:00:55.370
First recall the definition. So limit as x
approaches c of f(x) is a
 
0:00:55.370,0:01:01.629
number L; so c and L are both numbers, real
numbers. f is a function,
 
0:01:01.629,0:01:06.380
x is approaching c. And we said this is true
if the following -- for
 
0:01:06.380,0:01:10.180
every epsilon greater than zero, there exists
a delta greater than
 
0:01:10.180,0:01:14.800
zero such that for all x which are within delta
distance of c, f(x) is
 
0:01:14.800,0:01:17.590
within epsilon distance of L. Okay?
 
0:01:17.590,0:01:24.590
Now, how do we describe this in terms for
limit game?
 
0:01:26.530,0:01:33.530
KM: So, skeptic starts off with the first
part of the definition.
 
0:01:34.990,0:01:38.189
Vipul: By picking the epsilon? Okay, that's
the thing written in
 
0:01:38.189,0:01:42.939
black. What's the skeptic trying to do? What's the
goal of the skeptic?
 
0:01:42.939,0:01:49.100
KM: To try and pick an epsilon that would
not work.
 
0:01:49.100,0:01:53.450
Vipul: So the goal of the skeptic is to try
to show that the statement is false.
 
0:01:53.450,0:01:54.100
KM: Yeah.


==Conceptual definition and various cases==
0:01:54.100,0:01:57.790
Vipul: Right? In this case the skeptic should
try to start by choosing
 
0:01:57.790,0:02:02.220
an epsilon that is really [small] -- the goal of
the skeptic is to pick an
 
0:02:02.220,0:02:04.500
epsilon that's really small, what is the
skeptic trying to challenge
 
0:02:04.500,0:02:07.920
the prover into doing by picking the epsilon?
The skeptic is trying to


===Formulation of conceptual definition===
0:02:07.920,0:02:11.959
challenge the prover into trapping the function
close to L when x is


<center>{{#widget:YouTube|id=bE_aKfmUHN8}}</center>
0:02:11.959,0:02:17.040
close to c. And the way the skeptic specifies
what is meant by "close to L" is


===Functions of one variable case===
0:02:17.040,0:02:19.860
by the choice of epsilon. Okay?


This covers limits at and to infinity.
0:02:19.860,0:02:24.900
When picking epsilon the skeptic is
effectively picking this interval, L -
 
0:02:24.900,0:02:30.700
epsilon, L + epsilon). Okay? And basically
that's what the skeptic is
 
0:02:30.700,0:02:33.680
doing. The prover is then picking a delta.
What is the goal of the
 
0:02:33.680,0:02:36.239
prover in picking the delta? The prover is
saying, "Here's how I can
 
0:02:36.239,0:02:40.099
trap the function within that interval. I'm
going to pick a delta and
 
0:02:40.099,0:02:43.520
my claim is that if the x value within delta distance of c, except the
 
0:02:43.520,0:02:47.000
point c itself, so my claim is for any x value
there the function is
 
0:02:47.000,0:02:48.260
trapped in here."
 
0:02:48.260,0:02:52.819
So, the prover picks the delta and then the
skeptic tries to
 
0:02:52.819,0:02:56.709
test the prover's claim by picking an x
 
0:02:56.709,0:02:59.670
which is within the interval specified by
the prover and then they
 
0:02:59.670,0:03:03.379
both check whether f(x) is within epsilon
distance [of L]. If it is
 
0:03:03.379,0:03:07.940
then the prover wins and if it is not, if
this [|f(x) - L|]is not less
 
0:03:07.940,0:03:09.989
than epsilon then the skeptic wins. Okay?
 
0:03:09.989,0:03:13.659
So, the skeptic is picking the neighborhood
of the target point which
 
0:03:13.659,0:03:17.030
in this case is just the open interval of
radius epsilon, the prover
 
0:03:17.030,0:03:21.940
is picking the delta which is effectively the
neighborhood of the domain
 
0:03:21.940,0:03:25.760
point except the point c as I've said open
interval (c - delta, c +
 
0:03:25.760,0:03:30.870
delta) excluding c and then the skeptic picks
an x in the neighborhood
 
0:03:30.870,0:03:35.700
specified by prover and if the function value
is within the interval
 
0:03:35.700,0:03:38.830
specified by the skeptic then the prover wins.
 
0:03:38.830,0:03:41.989
Now, what does it mean to say the statement
is true in terms of the
 
0:03:41.989,0:03:43.080
game?
 
0:03:43.080,0:03:50.080
KM: So, it means that the prover is always
going to win the game.
 
0:03:51.849,0:03:55.629
Vipul: Well, sort of. I mean the prover may
play it stupidly. The
 
0:03:55.629,0:04:00.750
prover can win the game if the prover plays
well. So, the prover has a
 
0:04:00.750,0:04:03.230
winning strategy for the game. Okay?
 
0:04:05.230,0:04:10.299
The statement is true if the prover has a
winning strategy for the
 
0:04:10.299,0:04:14.090
game and that means the prover has a way
of playing the game such that
 
0:04:14.090,0:04:17.320
whatever the skeptic does the prover is going
to win the game. The
 
0:04:17.320,0:04:20.789
statement is considered false if the skeptic
has a winning strategy
 
0:04:20.789,0:04:23.370
for the game which means the skeptic has a
way of playing so that
 
0:04:23.370,0:04:25.729
whatever the prover does the skeptic can win
the game.
 
0:04:25.729,0:04:27.599
Or if the game doesn't make sense at all
...
 
0:04:27.599,0:04:29.460
maybe the function is not defined on
 
0:04:29.460,0:04:31.050
the immediate left and right of c.
 
0:04:31.050,0:04:32.370
If the function isn't defined then we
 
0:04:32.370,0:04:34.160
cannot even make sense of the statement.
 
0:04:34.160,0:04:36.990
Either way -- the skeptic has a winning strategy
 
0:04:36.990,0:04:37.770
or the game doesn't make sense --
 
0:04:41.770,0:04:43.470
then the statement is false.
 
0:04:43.470,0:04:47.660
If the prover has a winning strategy
the statement is true.
 
0:04:47.660,0:04:54.660
With this background in mind let's look
at some common misconceptions.
 
0:04:56.540,0:05:03.540
Okay. Let's say we are trying to prove that
the limit as x approaches
 
0:05:27.620,0:05:31.530
2 of x^2 is 4, so is that statement correct?
The statement we're
 
0:05:31.530,0:05:32.060
trying to prove?
 
0:05:32.060,0:05:32.680
KM: Yes.
 
0:05:32.680,0:05:35.960
Vipul: That's correct. Because in fact x^2
is a continuous function
 
0:05:35.960,0:05:40.160
and the limit of a continuous function at
the point is just the
 
0:05:40.160,0:05:43.030
value at the point and 2^2 is 4. But we're
going to now try to prove
 
0:05:43.030,0:05:48.530
this formally using the epsilon-delta definition
of limit, okay? Now
 
0:05:48.530,0:05:51.229
in terms of the epsilon-delta definition or
rather in terms of this
 
0:05:51.229,0:05:55.160
game setup, what we need to do is we need
to describe a winning
 
0:05:55.160,0:06:01.460
strategy for the prover. Okay? We need to
describe delta in terms of
 
0:06:01.460,0:06:05.240
epsilon. The prover essentially ... the only
move the prover makes is
 
0:06:05.240,0:06:09.130
this choice of delta. Right? The skeptic picked
epsilon, the prover
 
0:06:09.130,0:06:12.810
picked delta then the skeptic picks x and
then they judge who won. The
 
0:06:12.810,0:06:15.810
only choice the prover makes is the choice
of delta, right?
 
0:06:15.810,0:06:16.979
KM: Exactly.
 
0:06:16.979,0:06:20.080
Vipul: The prover has to specify delta in terms
of epsilon.
 
0:06:20.080,0:06:24.819
So, here is my strategy. My strategy is I'm
going to choose delta as,
 
0:06:24.819,0:06:29.509
I as a prover is going to choose delta as
epsilon over the absolute
 
0:06:29.509,0:06:33.690
value of x plus 2 [|x + 2|]. Okay?
 
0:06:33.690,0:06:36.880
Now, what I want to show that this strategy
works. So, what I'm claiming
 
0:06:36.880,0:06:39.840
is that if ... so let me just finish this
and then you can tell me where
 
0:06:39.840,0:06:43.419
I went wrong here, okay? I'm claiming that
this strategy works which
 
0:06:43.419,0:06:47.130
means I'm claiming that if the skeptic now
picks any x which is within
 
0:06:47.130,0:06:54.130
delta distance of 2; the target point,
 
0:06:56.710,0:07:01.490
then the function value is within epsilon
distance of 4, the claimed
 
0:07:01.490,0:07:04.080
limit. That's what I want to show.
 
0:07:04.080,0:07:08.300
Now is that true? Well, here's how I do
it. I say, I start by
 
0:07:08.300,0:07:13.539
taking this expression, I factor it as
|x - 2||x + 2|. The absolute
 
0:07:13.539,0:07:16.810
value of product is the product of the absolute
values so this can be
 
0:07:16.810,0:07:21.599
split like that. Now I say, well, we know
that |x - 2| is less than
 
0:07:21.599,0:07:24.979
delta and this is a positive thing. So we
can write this as less than delta
 
0:07:24.979,0:07:31.979
times absolute value x plus 2. Right? And
this delta is epsilon over
 
0:07:35.599,0:07:37.620
|x + 2| and we get epsilon.
 
0:07:37.620,0:07:40.460
So, this thing equals something, less than
something, equals
 
0:07:40.460,0:07:43.580
something, equals something, you have a chain
of things, there's one
 
0:07:43.580,0:07:47.720
step that you have less than. So overall we
get that this expression,
 
0:07:47.720,0:07:53.740
this thing is less than epsilon. So, we have
shown that whatever x the
 
0:07:53.740,0:08:00.370
skeptic would pick, the function value lies
within the epsilon
 
0:08:00.370,0:08:05.030
distance of the claimed limit. As long as the skeptic picks x within
 
0:08:05.030,0:08:09.240
delta distance of the target point.
 
0:08:09.240,0:08:16.240
Does this strategy work? Is this a proof?
What's wrong with this?
 
0:08:24.270,0:08:31.270
Do you think there's anything wrong
with the algebra I've done here?
 
0:08:33.510,0:08:40.510
KM: Well, we said that ...
 
0:08:40.910,0:08:47.910
Vipul: So, is there anything wrong in the
algebra here? This is this,
 
0:08:50.160,0:08:51.740
this is less than delta, delta ... So, this
part
 
0:08:51.740,0:08:52.089
seems fine, right?
 
0:08:52.089,0:08:52.339
KM: Yes.
 
0:08:52.330,0:08:55.640
Vipul: There's nothing wrong in the algebra
here. So, what could be
 
0:08:55.640,0:09:00.310
wrong? Our setup seems fine. If the x value
is within delta distance
 
0:09:00.310,0:09:03.350
of 2 then the function value is within epsilon
distance of 4. That's
 
0:09:03.350,0:09:05.360
exactly what we want to prove, right?
 
0:09:05.360,0:09:11.120
So, there's nothing wrong this point onward.
So, the error happened
 
0:09:11.120,0:09:14.440
somewhere here. What do you think
was wrong
 
0:09:14.440,0:09:21.160
here? In the strategy choice step? What do
you think went wrong in the
 
0:09:21.160,0:09:24.010
strategy choice step?
 
0:09:24.010,0:09:28.850
Well, okay, so in what order do they play their moves?
Skeptic will choose the epsilon,
 
0:09:28.850,0:09:29.760
then?
 
0:09:29.760,0:09:35.130
KM: Then the prover chooses delta.
 
0:09:35.130,0:09:36.080
Vipul: Prover chooses delta. Then?
 
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.
</toggledisplay>
 
==Conceptual definition and various cases==
 
===Formulation of conceptual definition===
 
<center>{{#widget:YouTube|id=bE_aKfmUHN8}}</center>
 
Full timed transcript: <toggledisplay>
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.</toggledisplay>
 
===Functions of one variable case===
 
This covers limits at and to infinity.


<center>{{#widget:YouTube|id=EOQby7b-WrA}}</center>
<center>{{#widget:YouTube|id=EOQby7b-WrA}}</center>
===Limit of sequence versus real-sense limit===
<center>{{#widget:YouTube|id=P9APtpIE4y8}}</center>
Full timed transcript: <toggledisplay>
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.</toggledisplay>


===Real-valued functions of multiple variables case===
===Real-valued functions of multiple variables case===


<center>{{#widget:YouTube|id=HZcYxcZplFA}}</center>
<center>{{#widget:YouTube|id=usb3jew_QVI}}</center>

Latest revision as of 22:29, 29 August 2013

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

The videos below are all taken from certain playlists. Instead of watching the videos on this page, you may prefer to watch the entire playlists on YouTube. Below are the playlist links:

Motivation and general idea

{{#widget:YouTube|id=iZ_fCNvYa9U}}

Full timed transcript: [SHOW MORE]

Definition for finite limit for function of one variable

Two-sided limit

{{#widget:YouTube|id=0vy0Fslxi-k}}

Full timed transcript: [SHOW MORE]

Left hand limit

Right hand limit

{{#widget:YouTube|id=qBjqc78KGx0}}

Full timed transcript: [SHOW MORE]

Relation between the limit notions

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

Two-sided limit

{{#widget:YouTube|id=Kh253PUghFk}}

Full timed transcript: [SHOW MORE]

{{#widget:YouTube|id=N0U8Y11nlPk}}

Full timed transcript: [SHOW MORE]

Non-existence of limit

{{#widget:YouTube|id=JoVuC4pksWs}}

Full timed transcript: [SHOW MORE]

Misconceptions

{{#widget:YouTube|id=F0r_offAc5M}}

Full timed transcript: [SHOW MORE]

Conceptual definition and various cases

Formulation of conceptual definition

{{#widget:YouTube|id=bE_aKfmUHN8}}

Full timed transcript: [SHOW MORE]

Functions of one variable case

This covers limits at and to infinity.

{{#widget:YouTube|id=EOQby7b-WrA}}

Limit of sequence versus real-sense limit

{{#widget:YouTube|id=P9APtpIE4y8}}

Full timed transcript: [SHOW MORE]

Real-valued functions of multiple variables case

{{#widget:YouTube|id=usb3jew_QVI}}
Retrieved from "https://calculus.subwiki.org/w/index.php?title=Video:Limit&oldid=2434"