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. That was just an intuitive idea, and a few somewhat

0:00:28.099,0:00:29.680 non-intuitive aspects of that.

0:00:29.680,0:00:36.680 Here I have the notation: "limit as x approaches c of f(x) is L" is

0:00:37.540,0:00:42.079 written like this. Limit ... Under the limit, we write where the

0:00:42.079,0:00:46.180 domain point goes, so x is approaching a value, c, and c could be an

0:00:46.180,0:00:51.059 actual number. x, however, will always be a variable letter. This x

0:00:51.059,0:00:54.519 will not be a number. c could be a number like zero, one, two, three,

0:00:54.519,0:00:55.329 or something.

0:00:55.329,0:01:02.050 f(x). f is the function. We are saying that as x approaches some

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

0:01:06.640,0:01:09.030 Limit as x approaches c of f(x) is L.

0:01:09.030,0:01:15.259 Now what does this mean? Roughly what it means is that as x is coming

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 its 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 a as a gap 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 food vault 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 of c, so this is the value x of c, and this is a 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 prospective 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 initial left of c, the value, Y

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

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

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 sides of limit, which

0:06:28.259,0:06:33.419 means that in this case the limit as x approaches c of f(x) does not

0:06:33.419,0:06:36.289 exist because you have a left-hand limit, and you have a right-hand

0:06:36.289,0:06:39.860 limit, and they are not equal to each other. The value, as such,

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

0:06:43.279,0:06:46.379 affect this concept of limit, but the real problem here is that the

0:06:46.379,0:06:48.490 left-hand limit and right-hand limit are not equal. The left-hand

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 they're headed to, and add the Y

0:07:21.129,0:07:22.240 coordinate of that.

0:07:22.240,0:07:29.240 Let me make an example, where the limit does exist. Let's say you

0:07:42.899,0:07:48.449 have a picture, something like this. In this case, the left-hand limit

0:07:48.449,0:07:52.610 and right-hand limit are the same thing, so this number, but the

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

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

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

0:08:03.139,0:08:04.719 limit are the same.

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

0:08:09.979,0:08:13.710 idea, which is called the two-finger test. You may have heard it in

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 where you’ve done

0:08:59.819,0:09:05.779 limits in detail in school. However, what I want to say here is that

0:09:05.779,0:09:11.850 this two-finger test is not really a good definition of limit. What’s

0:09:11.850,0:09:13.600 the problem? The problem is that you could have really crazy

0:09:13.600,0:09:18.790 function, and it's really hard to move your finger along the graph of

0:09:18.790,0:09:25.220 the function. If the function sort of jumps around a lot, it's really

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

0:09:29.440,0:09:35.100 mathematically pure thing. It's like trying to answer the

0:09:35.100,0:09:39.540 mathematical question using a physical description, which is sort of

0:09:39.540,0:09:41.579 the wrong type of answer.

0:09:41.579,0:09:45.610 While this is very good for a basic intuition for very simple types of

0:09:45.610,0:09:50.040 functions, it's not actually the correct idea of limit. What kind of

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

0:09:56.990,0:10:03.209 understanding of limit? The main thing is functions which have a lot

0:10:03.209,0:10:07.980 of oscillation. Let me do an example.

0:10:07.980,0:10:14.980 I'm now going to write down a type of function where, in fact, you

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

0:10:21.899,0:10:28.899 question precisely. This is a graph of a function, sine 1 over x.

0:10:28.959,0:10:32.920 Now this looks a little weird. It's not 1 over sine x; that would

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

0:10:44.879,0:10:50.220 function itself is not defined at x equals zero, but just the fact

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

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

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

0:10:57.660,0:11:04.660 Let's first make the picture of sine x. Sine-x looks like that. How

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

0:11:20.100,0:11:25.759 When x is very large positive, 1 over x is near zero, slightly

0:11:25.759,0:11:30.660 positive, just slightly bigger than zero, and sine 1 over x is

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

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

0:11:42.810,0:11:49.420 this path, but much more slowly, each one, then it's going to go this

0:11:49.420,0:11:56.420 path, but in reverse, so like that. Then it's going to go this path,

0:11:57.149,0:12:00.740 but now it does all these oscillations, all of these oscillations. It

0:12:00.740,0:12:03.569 has to go faster and faster.

0:12:03.569,0:12:10.569 For instance, this is pi, this 1 over pi, then this is 2 pi, this

0:12:12.329,0:12:16.990 number is 1 over 2 pi, then the then next time it reaches zero will be

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

0:12:21.160,0:12:24.579 going to be crazily oscillating between minus 1, and 1. The frequency

0:12:24.579,0:12:29.170 of the oscillation keeps getting faster and faster as you come closer

0:12:29.170,0:12:34.050 and closer to zero. The same type of picture on the left side as

0:12:34.050,0:12:40.360 well; it's just that it's an odd function. It's this kind of picture.

0:12:40.360,0:12:47.360 I'll make a bigger picture here ... I'll make a bigger picture on another

0:12:53.649,0:13:00.649 one. all of these oscillation should be between minus 1 and 1, and we

0:13:22.439,0:13:29.399 get faster so we get faster and faster, and now my pen is too thick.

0:13:29.399,0:13:31.600 It's the same, even if you used your finger instead of the pen to

0:13:31.600,0:13:38.600 place it, it would be too thick, it's called the thick finger problem.

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

0:13:45.060,0:13:49.199 finger is too thick, but actually, there's a very thin line, and it's

0:13:49.199,0:13:52.519 an infinitely thin line of the graph, which goes like that.

0:13:52.519,0:13:59.519 Let's get back to our question: What is limit as x approaches zero,

0:14:02.699,0:14:09.699 sine 1 over x. I want you to think about this a bit. Think about like

0:14:13.439,0:14:18.050 the finger test. You move your finger around, move it like this,

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

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

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

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

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

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

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

0:14:52.949,0:14:59.949 coming close to zero. That any number of the form x is 1 over N pi,

0:15:00.329,0:15:07.329 sine 1 over x is zero. It keeps coming close to zero. As x

0:15:07.990,0:15:12.459 approaches zero, this number keeps coming close to zero.

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

0:15:17.449,0:15:24.449 approaches zero, sine 1 over x is sort of coming close to zero, is it?

0:15:31.230,0:15:36.550 It's definitely coming near zero, right? Anything you make around

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

0:15:41.920,0:15:42.399 enter that.

0:15:42.399,0:15:47.269 On the other hand, it's not really staying close to zero. It's kind of

0:15:47.269,0:15:50.300 oscillating with the minus 1 and 1. However, smaller interval you

0:15:50.300,0:15:54.540 take around zero on the x thing, the function is oscillating between

0:15:54.540,0:15:57.600 minus 1 and 1. It's not staying faithful to zero.

0:15:57.600,0:16:02.249 Now you have kind of this question: What should be the correct

0:16:02.249,0:16:09.249 definition of this limit? Should it mean that it approaches the

0:16:10.029,0:16:15.100 point, but maybe goes in and out, close and far? Or should it mean it

0:16:15.100,0:16:18.879 approaches and stays close to the point? That is like a judgment you

0:16:18.879,0:16:22.629 have to make in the definition, and it so happens that people who

0:16:22.629,0:16:28.639 tried defining this chose the latter idea; that is, it should come

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

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

0:16:38.290,0:16:43.639 function needs to be trapped near the limit, close to the point in the

0:16:43.639,0:16:45.079 domain.

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

0:16:49.459,0:16:54.420 function is oscillating too widely. You cannot trap it. You cannot

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

0:17:01.059,0:17:08.059 function value, say, in this small horizontal strip near zero. You

0:17:08.319,0:17:11.650 cannot trap in the area, so that means the limit cannot be zero, but

0:17:11.650,0:17:15.400 the same logic works anywhere else. The limit cannot be half, because

0:17:15.400,0:17:20.440 you cannot trap the function in a small horizontal strip about half

0:17:20.440,0:17:22.130 whereas x approaches zero.

0:17:22.130,0:17:26.440 We will actually talk about this example in great detail in our future

0:17:26.440,0:17:30.330 with you after we've seen the formal definition, but the key idea you

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

0:17:33.890,0:17:37.340 to the point of its limit. It actually needs to stay close. It needs

0:17:37.340,0:17:41.050 to be trapped near the point.

0:17:41.050,0:17:44.810 The other important idea regarding limits is that the limit depends

0:17:44.810,0:17:50.370 only on the behavior very, very close to the point. What do I mean by

0:17:50.370,0:17:56.580 very, very close? If you were working it like, the real goal, you may

0:17:56.580,0:18:02.300 say, it's like, think of some really small number and you say that

0:18:02.300,0:18:07.050 much distance from it. Let's say I want to get the limit as x

0:18:07.050,0:18:14.050 approaches 2...I'll just write it here. I want to get, let's say,

0:18:23.520,0:18:30.520 limit has x approaches 2 of some function, we may say, well, we sort

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

0:18:38.750,0:18:43.380 What about 2.0000001? Is that close enough?

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

0:18:47.420,0:18:54.420 this is close enough." The difference is like ... so it's

0:18:57.040,0:19:04.040 10^{-7}. It's really only close to 2 compared to our usual sense of

0:19:12.990,0:19:16.670 numbers, but as far as mathematics is concerned, both of these numbers

0:19:16.670,0:19:21.110 are really far from 2. Any individual number that is not 2 is very

0:19:21.110,0:19:22.130 far from 2.

0:19:22.130,0:19:29.130 What do I mean by that, well, think back to one of our

0:19:29.670,0:19:36.670 pictures. Here's a picture. Supposed I take some points. Let's say

0:19:41.970,0:19:47.640 this is 2, and suppose I take one point here, which is really close to

0:19:47.640,0:19:50.970 2, and I just change the value of the function at that point. I

0:19:50.970,0:19:55.200 change the value of the function at that point, or I just change the

0:19:55.200,0:19:59.990 entire picture of the graph from that point rightward. I just take

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

0:20:05.940,0:20:11.410 picture by that picture, or I replace the picture by some totally new

0:20:11.410,0:20:15.250 picture like that picture. I just change the part of the graph to the

0:20:15.250,0:20:21.440 right of some point, like 2.00001, whatever. Will that effect the

0:20:21.440,0:20:25.770 limit at 2? No, because the limit at 2 really depends only on the

0:20:25.770,0:20:27.520 behavior if you're really, really close.

0:20:27.520,0:20:32.040 If you take any fixed point, which is not 2, and you change the

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

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

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

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

0:20:52.060,0:20:56.570 The limit depends on the behavior arbitrarily close to the point. It

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

0:21:00.210,0:21:06.910 just depends on the behavior as you approach the point and any other

0:21:06.910,0:21:11.330 point is far away. It's only sort of together that all the other

0:21:11.330,0:21:16.230 points matter, and it's only them getting really close that

0:21:16.230,0:21:19.790 matters. The other thing is that the function actually needs to be

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

0:21:26.860,0:21:29.650 picture where it's oscillating between minus 1 and 1, however close

0:21:29.650,0:21:35.150 you get to zero, keeps oscillating, and so you cannot trap it around

0:21:35.150,0:21:40.590 any point. You cannot trap the function value in any small enough

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

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

0:21:54.630,0:22:00.640 then we'll come back to some of these issues later with the formal

0:22:00.640,0:22:01.870

understanding.

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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Non-existence of limit

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Misconceptions

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Conceptual definition and various cases

Formulation of conceptual definition

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Functions of one variable case

This covers limits at and to infinity.

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Real-valued functions of multiple variables case

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