Video:Limit

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Motivation and general idea

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Full timed transcript: [SHOW MORE]

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. 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 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 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.

Definition for finite limit for function of one variable

Two-sided limit

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Full timed transcript: [SHOW MORE]

0:00:15.809,0:00:20.490 Vipul: In this talk, I'm going to introduce the definition, the formal epsilon delta definition

0:00:20.490,0:00:24.669 of a two-sided limit for a function of a one variable, that's called f.

0:00:24.669,0:00:31.349 I'm going to assume there is a point c and c doesn't actually have to be in the domain of f.

0:00:31.349,0:00:38.030 Thus f doesn't have to be defined at c for this notion to make sense rather f is defined around c.

0:00:38.030,0:00:44.909 What that means is f is defined on some open set containing c.

0:00:51.009,0:01:03.009 Let's make a picture here so you have c, c + t, c -- t.

0:01:03.040,0:01:11.040 What this is saying is there is some t probably small enough so that the function is defined

0:01:12.549,0:01:18.590 in here and may be it's not defined at the point c.

0:01:18.590,0:01:31.590 This set for some t>0. The function is defined on the immediate left of c and it is defined

0:01:31.999,0:01:34.770 on the immediate right of c.

0:01:34.770,0:01:38.890 We need that in order to make sense of what I'm going to say.

0:01:38.890,0:01:44.590 We say that limit as x approaches c of f(x) is L where L is some other real number or

0:01:44.590,0:01:49.679 maybe it's the same real number [as c], so we say this limit equals L, now I'll write the definition

0:01:49.679,0:01:56.679 in multiple lines just to be clear about the parts of the definition.

0:01:56.770,0:02:39.770 For every epsilon > 0. This is epsilon. There exists delta > 0 such that for all x in R satisfying...what?

0:02:41.070,0:02:45.070 Rui: Satisfying |x -- c| ...

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.

Right hand limit

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Full timed transcript: [SHOW MORE]

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!]

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

Two-sided limit

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Full timed transcript: [SHOW MORE]

0:00:15.589,0:00:21.160 Vipul: In this video, I'm going to go over the usual definition of limit and think of

0:00:21.160,0:00:24.930 it in terms of a game.

0:00:24.930,0:00:26.390 The game is as follows.

0:00:26.390,0:00:27.340 Consider this statement.

0:00:27.340,0:00:31.509 You are saying limit as x approaches c of f(x) is L.

0:00:31.509,0:00:32.029 Okay.

0:00:32.029,0:00:35.160 There are two players to this game.

0:00:35.160,0:00:38.600 One is the prover and one is the skeptic.

0:00:38.600,0:00:44.550 The prover's goal is to show that this claim is true so the prover is trying to convince

0:00:44.550,0:00:48.730 the skeptic that this limit as x approaches c of f(x) is L,

0:00:48.730,0:01:01.160 the skeptic will try to ask tough questions and see if the prover can still manage to show this.

0:01:01.160,0:01:04.059 The way the game is structured is as follows.

0:01:04.059,0:01:08.899 Let me just go over the individual components of the statement for the limit and I will

0:01:08.899,0:01:10.610 translate each one.

0:01:10.610,0:01:17.610 I will explain the game and then explain how it corresponds to the definition you've seen.

0:01:20.219,0:01:27.219 We begin with the skeptic chooses epsilon > 0.

0:01:35.840,0:01:42.840 This is the part of the definition which reads for every epsilon > 0.

0:01:47.099,0:01:53.289 That's the first clause of the definition and that's basically the skeptic is choosing

0:01:53.289,0:01:54.579 epsilon > 0.

0:01:54.579,0:01:59.299 What is the skeptic trying to do when choosing epsilon > 0?

0:01:59.299,0:02:06.299 What the skeptic is effectively doing is choosing this interval L -- epsilon to L + epsilon.

0:02:14.400,0:02:18.220 The skeptic is effectively trying to choose this interval L -- epsilon to L + epsilon.

0:02:18.220,0:02:26.110 What is the skeptic trying the challenge the prover into doing when picking this interval? [ANSWER!]

0:02:26.110,0:02:29.890 Rui: Whether the prover can trap.

0:02:29.890,0:02:35.180 Vipul: The skeptic is trying to challenge (and this will become a clearer a little later).

0:02:35.180,0:02:41.790 The idea is, the skeptic is trying to challenge the prover into trapping the function when

0:02:41.790,0:02:47.620 the input x is close to c, trapping the function output within this interval and that's

0:02:47.620,0:02:52.459 not clear which is why we need to continue its definition.

0:02:52.459,0:02:58.609 The prover chooses. What does the prover choose? [ANSWER!]

0:02:58.609,0:03:00.260 Rui: delta.

0:03:00.260,0:03:07.260 Vipul: delta > 0 and this corresponds to the next part of the definition which says

0:03:08.480,0:03:15.480 there exists delta > 0.

0:03:19.749,0:03:26.749 In this picture, which I have up here, this is the value c.

0:03:28.840,0:03:31.989 This is c + delta and this is c -- delta.

0:03:31.989,0:03:41.349 This is c and L, so c is the x coordinate, L is the function value or limited the function value.

0:03:41.349,0:03:48.349 The skeptic chooses this strip like this from L -- epsilon to L + epsilon by choosing epsilon

0:03:51.450,0:03:56.109 so the skeptic just chooses the number absent what it is effectively doing is to choose

0:03:56.109,0:04:01.790 this strip, L -- epsilon to L + epsilon. The prover then chooses a delta.

0:04:01.790,0:04:03.829 What's the prover effectively choosing?

0:04:03.829,0:04:07.290 The prover is effectively choosing this interval.

0:04:07.290,0:04:14.230 Okay so that's this interval.

0:04:14.230,0:04:20.209 It is c -- delta to c + delta except you don't really care about the point c itself,

0:04:20.209,0:04:26.490 (but that's a little subtlety we don't have to bother about), so the skeptic is choosing

0:04:26.490,0:04:29.780 the interval like this. The prover is choosing the interval like this.

0:04:29.780,0:04:33.340 How is the skeptic choosing the interval? By just specifying the value of epsilon.

0:04:33.340,0:04:34.880 How is the prover choosing [the interval around c]?

0:04:34.880,0:04:45.880 By just specifying a value of delta. Okay. Now what does the skeptic now do? [ANSWER!]

0:04:46.500,0:04:52.979 Rui: Skeptic will check.

0:04:53.079,0:05:00.079 Vipul: There is something more to choose (right?) before checking.

0:05:02.710,0:05:06.599 What does the definition say? For every epsilon > 0 there exists a delta greater than zero

0:05:06.599,0:05:07.259 such that ... [COMPLETE!]

0:05:07.259,0:05:08.580 Rui: For every.

0:05:08.580,0:05:13.220 Vipul: For every x such that something. The skeptic can now pick x.

0:05:13.220,0:05:17.000 Rui: That's what I meant by checking.

0:05:17.000,0:05:21.940 Vipul: The skeptic could still, like, pick a value to challenge the prover.

0:05:21.940,0:05:28.940 The skeptic chooses x but what x can the skeptic choose?

0:05:29.169,0:05:31.810 Rui: Within the...

0:05:31.810,0:05:36.590 Vipul: This interval which the prover has specified.

0:05:36.590,0:05:43.590 The skeptic is constrained to choose x within the interval.

0:05:44.250,0:05:49.639 That's the same as c -- delta ... Is this all coming?

0:05:49.639,0:05:50.330 Rui: Yes.

0:05:50.330,0:05:57.330 Vipul: c -- delta, c union c to c + delta.

0:05:59.110,0:06:15.110 The way it's written is for every x in this interval.

0:06:16.849,0:06:21.349 Lot of people write this in a slightly different way.

0:06:21.349,0:06:28.349 They write it as ...

0:06:28.400,0:06:31.720 (You should see the definition video before this.)

0:06:31.720,0:06:37.729 (I'm sort of assuming that you have seen the definition -- this part [of the screen] so you can map it)

0:06:37.729,0:06:40.000 so a lot of people write it like this.

0:06:40.000,0:06:45.190 It is just saying x is within delta distance of c but it's not equal to c itself.

0:06:45.190,0:06:50.949 Now it's time for the judge to come in and decide who has won.

0:06:50.949,0:06:55.930 How does the judge decide? [ANSWER!]

0:06:55.930,0:07:01.360 Rui: For the x that the skeptic chooses and see the corresponding y.

0:07:01.360,0:07:03.289 Vipul: The f(x) value.

0:07:03.289,0:07:10.289 Rui: If the f(x) value is within the horizontal strip then the prover wins.

0:07:12.509,0:07:30.000 Vipul: If |f(x) -- L| < epsilon which is the same as saying f(x) is in what interval? [ANSWER!]

0:07:30.000,0:07:41.620 L- epsilon to L + epsilon then the prover wins. Otherwise? [ANSWER!]

0:07:42.120,0:07:46.120 Rui: The skeptic wins.

0:07:46.120,0:07:53.120 [But] the skeptic can choose a really dumb [stupid] x.

0:07:54.039,0:07:57.610 Vipul: That's actually the next question I want to ask you.

0:07:57.610,0:08:01.240 What does it actually mean to say that this statement is true?

0:08:01.240,0:08:04.770 Is it just enough that the prover wins? That's not enough.

0:08:04.770,0:08:07.909 What do you want to say to say that this statement is true?

0:08:07.909,0:08:11.210 Rui: For every x in the interval.

0:08:11.210,0:08:16.289 Vipul: For every x but not only for every x you should also say for every epsilon.

0:08:16.289,0:08:22.139 All the moves that the skeptic makes, the prover should have a strategy, which works for all of them.

0:08:22.139,0:08:25.710 So, this statement is true [if] ...

0:08:25.710,0:08:29.800 This is true if the prover has what for the game? [ANSWER!]

0:08:30.539,0:08:35.050 Rui: Winning strategy. Vipul: Winning what? Rui: Strategy.

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.

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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

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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.

Misconceptions

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

Full timed transcript: [SHOW MORE]

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.

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

0:02:07.920,0:02:11.959 challenge the prover into trapping the function close to L when x is

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

0:02:17.040,0:02:19.860 by the choice of epsilon. Okay?

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 be right "for every" "for all." Right? And

0:19:04.260,0:19:07.390 for all the moves of the prover it's "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.

Conceptual definition and various cases

Formulation of conceptual definition

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Full timed transcript: [SHOW MORE]

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

This covers limits at and to infinity.

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Limit of sequence versus real-sense limit

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Full timed transcript: [SHOW MORE]

0:00:15.530,0:00:21.640 Vipul: Okay. So this talk is going to be about limit at infinity for

0:00:21.640,0:00:27.000 functions on real numbers and the concept of limits of sequences, how

0:00:27.000,0:00:33.320 these definitions are essentially almost the same thing and how they differ.

0:00:34.780,0:00:40.800 Okay. So let's begin by reviewing the definition of the limit as x

0:00:40.800,0:00:43.950 approaches infinity of f(x). Or rather what it means for that limit to

0:00:43.950,0:00:49.050 be a number L. Well, what it means is that for every epsilon greater

0:00:49.050,0:00:52.699 than zero, so we first say for every neighborhood of L, small

0:00:52.699,0:00:58.620 neighborhood of L, given by radius epsilon there exists a neighborhood

0:00:58.620,0:01:03.010 of infinity which is specified by choosing some a such that that is

0:01:03.010,0:01:08.670 the interval (a,infinity) ...

0:01:08.670,0:01:14.110 ... such that for all x in the interval from a to infinity. That is

0:01:14.110,0:01:18.770 for all x within the chosen neighborhood of infinity, the f(x) value

0:01:18.770,0:01:23.390 is within the chosen neighborhood of L. Okay?

0:01:23.390,0:01:27.290 If you want to think about it in terms of the game between the prover

0:01:27.290,0:01:32.090 and the skeptic, the prover is claiming that the limit as x approaches

0:01:32.090,0:01:36.079 infinity of f(x) is L. The skeptic begins by picking a neighborhood of

0:01:36.079,0:01:39.619 L which is parameterized by its radius epsilon. The prover picks the

0:01:39.619,0:01:42.939 neighborhood of infinity which is parameterized by its lower end

0:01:42.939,0:01:48.350 a. Then the skeptic picks a value x between a and infinity. Then they

0:01:48.350,0:01:51.530 check whether absolute value f(x) minus L [symbolically: |f(x) - L|] is

0:01:51.530,0:01:53.700 less than epsilon. That is they check whether f(x) is in the chosen

0:01:53.700,0:01:58.729 neighborhood of L (the neighborhood chosen by the skeptic). If it is,

0:01:58.729,0:02:03.359 then the prover wins. The prover has managed to trap the function: for

0:02:03.359,0:02:05.909 x large enough, the prover has managed to trap the function within

0:02:05.909,0:02:12.750 epsilon distance of L. If not, then the skeptic wins. The statement

0:02:12.750,0:02:18.680 is true if the prover has a winning the strategy for the game.

0:02:18.680,0:02:20.870 Now, there is a similar definition which one has for sequences. So,

0:02:20.870,0:02:24.390 what's a sequence? Well, it's just a function from the natural

0:02:24.390,0:02:27.319 numbers. And, here, we're talking of sequences of real numbers. So,

0:02:27.319,0:02:31.610 it's a function from the naturals to the reals and we use the same

0:02:31.610,0:02:34.950 letter f for a good reason. Usually we write sequences with

0:02:34.950,0:02:39.030 subscripts, a_n type of thing. But I'm using it as a function just to

0:02:39.030,0:02:43.159 highlight the similarities. So, limit as n approaches infinity, n

0:02:43.159,0:02:47.019 restricted to the natural numbers ... Usually if it's clear we're

0:02:47.019,0:02:52.260 talking of a sequence, we can remove this part [pointing to the n in N

0:02:52.260,0:02:53.700 constraint specification] just say limit n approaches infinity f(n),

0:02:53.700,0:02:57.220 but since we want to be really clear here, I have put this line. Okay?

0:02:57.220,0:03:01.090 So, this limit equals L means "for every epsilon greater than 0 ..."

0:03:01.090,0:03:05.159 So, it starts in the same way. The skeptic picks a neighborhood of

0:03:05.159,0:03:09.989 L. Then the next line is a little different but that's not really the

0:03:09.989,0:03:16.989 crucial part. The skeptic is choosing epsilon. The prover picks n_0, a

0:03:18.920,0:03:22.209 natural number. Now, here the prover is picking a real number. Here

0:03:22.209,0:03:25.010 the prover is picking a natural number. That's not really the big

0:03:25.010,0:03:30.310 issue. You could in fact change this line to match. You could

0:03:30.310,0:03:33.659 interchange these lines. It wouldn't affect either definition.

0:03:33.659,0:03:39.430 The next line is the really important one which is different. In here

0:03:39.430,0:03:43.680 [pointing to real-sense limit], the condition has to be valid for all

0:03:43.680,0:03:48.879 x, for all real numbers x which are bigger than the threshold which

0:03:48.879,0:03:51.810 the prover has chosen. Here on the other hand [pointing to the

0:03:51.810,0:03:54.909 sequence limit] the condition has to be valid for all natural numbers

0:03:54.909,0:03:58.480 which are bigger than the threshold the prover has chosen. By the way,

0:03:58.480,0:04:02.769 some of you may have seen the definition with an equality sign

0:04:02.769,0:04:08.439 here. It doesn't make a difference to the definition. It does affect

0:04:08.439,0:04:12.019 what n_0 you can choose, it will go up or down by one, but that's not

0:04:12.019,0:04:16.060 really a big issue. The big issue, the big difference between these

0:04:16.060,0:04:20.030 two definitions is that in this definition you are insisting that the

0:04:20.030,0:04:25.880 condition here is valid for all real x. So, you are insisting or

0:04:25.880,0:04:32.440 rather the game is forcing the prover to figure out how to trap the

0:04:32.440,0:04:36.090 function values for all real x. Whereas here, the game is only

0:04:36.090,0:04:39.639 requiring the prover to trap the function values for all large enough

0:04:39.639,0:04:42.080 natural numbers. So, here [real-sense limit] it's all large enough

0:04:42.080,0:04:44.780 real numbers. Here [sequence limit] it's all large enough natural

0:04:44.780,0:04:49.250 numbers. Okay?

0:04:49.250,0:04:56.250 So, that's the only difference essentially. Now, you can see from the

0:04:56.350,0:04:58.100 way we have written this that this [real-sense limit] is much

0:04:58.100,0:05:01.360 stronger. So, if you do have a function which is defined on real so

0:05:01.360,0:05:06.410 that both of these concepts can be discussed. If it were just a

0:05:06.410,0:05:08.870 sequence and there were no function to talk about then obviously, we

0:05:08.870,0:05:12.220 can't even talk about this. If there's a function defined on the reals

0:05:12.220,0:05:18.669 or on all large enough reals, then we can try taking both of

0:05:18.669,0:05:21.470 these. The existence of this [pointing at the real-sense limit] and

0:05:21.470,0:05:24.150 [said "or", meant "and"] it's being equal to L as much stronger than

0:05:24.150,0:05:26.550 this [the sequence limit] equal to L. If this is equal to L then

0:05:26.550,0:05:29.330 definitely this [the sequence limit] is equal to L. Okay?

0:05:29.330,0:05:31.139 But maybe there are situations where this [the sequence limit] is

0:05:31.139,0:05:35.330 equal to some number but this thing [the real-sense limit] doesn't

0:05:35.330,0:05:39.180 exist. So, I want to take one example here. I have written down an

0:05:39.180,0:05:46.180 example and we can talk a bit about that is this. So, here is a

0:05:46.389,0:05:53.389 function. f(x) = sin(pi x). This is sin (pi x) and the corresponding

0:05:55.630,0:05:58.930 function if you just restrict [it] to the natural numbers is just sin

0:05:58.930,0:06:04.440 (pi n). Now, what does sin (pi n) look like for a natural number then?

0:06:04.440,0:06:09.810 In fact for any integer n? pi times n is an integer multiple of

0:06:09.810,0:06:16.810 pi. sin of integer multiples of pi is zero. Let's make a picture of sin ...

0:06:22.830,0:06:27.289

0:06:27.289,0:06:31.800 It's oscillating. Right? Integer multiples of pi are precisely the

0:06:31.800,0:06:38.800 ones where it's meeting the axis. So, in fact we are concerned about

0:06:39.750,0:06:41.300 the positive one because we are talking of the sequence (natural

0:06:41.300,0:06:45.840 number [inputs]). Okay? And so, if you are looking at this sequence,

0:06:45.840,0:06:50.810 all the terms here are zero. So, the limit is also zero. So, this

0:06:50.810,0:06:53.030 limit [the sequence limit] is zero.

0:06:53.030,0:07:00.030 Okay. What about this limit? Well, we have the picture again. Is it

0:07:03.110,0:07:07.270 going anywhere? No. It's oscillating between minus one and one

0:07:07.270,0:07:08.970 [symbolically: oscillating in [-1,1]]. It's not settling down to any

0:07:08.970,0:07:11.669 number. It's not... You cannot trap it near any particular number

0:07:11.669,0:07:15.970 because it's all over the map between minus one and one. For the same

0:07:15.970,0:07:19.610 reason that sin(1/x) doesn't approach anything as x approaches zero,

0:07:19.610,0:07:23.949 the same reason sin x or sin(pi x) doesn't approach anything as x

0:07:23.949,0:07:30.949 approaches infinity. So, the limit for the real thing, this does not

0:07:31.419,0:07:37.370 exist. So, this gives an example where the real thing [the real-sense

0:07:37.370,0:07:41.590 limit] doesn't exist and the sequence thing [sequence limit] does

0:07:41.590,0:07:45.580 exist and so here is the overall summary. If the real sense limit,

0:07:45.580,0:07:47.430 that is this one [pointing to definition of real sense limit] exists,

0:07:47.430,0:07:51.150 [then] the sequence limit also exists and they're both equal. On the

0:07:51.150,0:07:53.819 other hand, you can have a situation with the real sense limit, the

0:07:53.819,0:07:57.440 limit for the function of reals doesn't exist but the sequence limit

0:07:57.440,0:08:00.819 still exists like this set up. Right?

0:08:00.819,0:08:03.590 Now, there is a little caveat that I want to add. If the real sense

0:08:03.590,0:08:08.729 limit doesn't exist as a finite number but it's say plus infinity then

0:08:08.729,0:08:12.009 the sequence limit also has to be plus infinity. If the real sense

0:08:12.009,0:08:16.990 limit is minus infinity, then the sequence limit also has to be minus

0:08:16.990,0:08:19.740 infinity. So, this type of situation, where the real sense limit

0:08:19.740,0:08:24.539 doesn't exist but the sequence exists, well, will happen in kind of

0:08:24.539,0:08:28.169 oscillatory type of situations. Where the real sense you have an

0:08:28.169,0:08:31.639 oscillating thing and in the sequence thing on the other hand you

0:08:31.639,0:08:34.570 somehow manage to pick a bunch of points where that oscillation

0:08:34.570,0:08:36.789 doesn't create a problem. Okay?

0:08:36.789,0:08:42.780 Now, why is this important? Well, it's important because in a lot of

0:08:42.780,0:08:47.120 cases when you have to calculate limits of sequences, you just

0:08:47.120,0:08:54.120 calculate them by doing, essentially, just calculating the limits of the

0:08:54.640,0:08:59.190 function defining the sequence as a limit of a real valued

0:08:59.190,0:09:03.460 function. Okay? 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)

0:09:10.510,0:09:17.510 or something like that. Right? Some rational function.

0:09:21.720,0:09:25.980 You just do this calculation as if you were just doing a

0:09:25.980,0:09:29.430 limit of a real function, function of real numbers, right? The answer

0:09:29.430,0:09:32.100 you get will be the correct one. If it's a finite number it will be

0:09:32.100,0:09:35.790 the same finite number. In this case it will just be one. But any

0:09:35.790,0:09:38.840 rational function, if the answer is finite, same answer for the

0:09:38.840,0:09:44.070 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.

0:09:46.250,0:09:51.420 However, if the answer you get for the real-sense limit is oscillatory

0:09:51.420,0:09:57.540 type of non existence, then that's inconclusive as far as the sequence

0:09:57.540,0:10:00.410 is concerned. You actually have to think about the sequence case and

0:10:00.410,0:10:05.700 figure out for yourself what happens to the limit. Okay? If might in

0:10:05.700,0:10:08.040 fact be the case that the sequence limit actually does exist even

0:10:08.040,0:10:11.380 though the real sense [limit] is oscillatory.

Okay.

Real-valued functions of multiple variables case

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