1 00:00:00,000 --> 00:00:09,285 [SOUND] So here's a table of values of the function F of X equals sin 1 / X. 2 00:00:09,285 --> 00:00:14,100 F of one is really sin of one, it's like 8.. 3 00:00:14,100 --> 00:00:21,322 F of 1. which is really sin of one over 1. sin of 10 -.5. 4 00:00:21,322 --> 00:00:25,030 F of 01.. This will be sine of 100, it's also about 5 00:00:25,030 --> 00:00:27,889 -0.5. F of 0.001, which is like sine of 1000, 6 00:00:27,889 --> 00:00:33,062 well that's 0.8 and some more, right? So the question is these numbers aren't 7 00:00:33,062 --> 00:00:36,261 really getting close to anything in particular. 8 00:00:36,261 --> 00:00:41,639 Can you really say that if you evaluate f at values which are close to but not 9 00:00:41,639 --> 00:00:46,608 equal to zero, that the outputs are actually getting close to anything in 10 00:00:46,608 --> 00:00:49,603 particular. I mean this is positive, negative, 11 00:00:49,603 --> 00:00:54,300 negative, positive, negative, positive, negative, it's looking pretty bad. 12 00:00:55,680 --> 00:00:58,227 Instead of a table, let's look at a graph. 13 00:00:58,227 --> 00:01:02,391 Here, I've got a graph of the funtcion f (x) = sin 1 / x. 14 00:01:02,391 --> 00:01:06,741 And you see the middle of this graph is just that horrible green blob. 15 00:01:06,741 --> 00:01:09,860 Right? It's really hard to make out any detail. 16 00:01:09,860 --> 00:01:14,701 You might think that's just a consequence of the fact that I'm drawing this graph 17 00:01:14,701 --> 00:01:18,703 with such thick lines. You know, and if I used thinner lines to 18 00:01:18,703 --> 00:01:24,180 draw my graph, maybe I could, you know get rid of this green blob and really see 19 00:01:24,180 --> 00:01:27,924 some detail. Even if I dial down the size of the lines 20 00:01:27,924 --> 00:01:32,984 that I'm using to draw this graph, the blob thing is still there, you know. 21 00:01:32,984 --> 00:01:36,520 And it's really there in the graph of the function. 22 00:01:36,520 --> 00:01:42,205 Even if these lines were true lines, zero thickness, it wouldn't be possible to fit 23 00:01:42,205 --> 00:01:47,266 even a single atom next to the Y axis without touching the graph of this 24 00:01:47,266 --> 00:01:50,649 function. The graph is oscillating wildly near 25 00:01:50,649 --> 00:01:54,080 zero. Even if your input is very close to zero 26 00:01:54,080 --> 00:01:57,810 your output could be anything between -1 and 1. 27 00:01:57,810 --> 00:02:08,762 So in light of this evidence, the limit of sine 1 / x as x approaches 0 does not. 28 00:02:08,762 --> 00:02:14,643 [SOUND] exist. Which sometimes I'll abbreviate DNE, for 29 00:02:14,643 --> 00:02:18,676 does not exist. what does it even mean to say it doesn't 30 00:02:18,676 --> 00:02:20,970 exist? What do we mean by the definition of 31 00:02:20,970 --> 00:02:23,814 limit? To say the limit equals something means 32 00:02:23,814 --> 00:02:28,484 that I can make the output as close as I want to l by making x close to a. 33 00:02:28,484 --> 00:02:33,722 So when I say this limit doesn't exist, I mean it's not the case that this limit is 34 00:02:33,722 --> 00:02:37,319 equal to anything, okay? If you tell me this limit is some 35 00:02:37,319 --> 00:02:41,089 positive number, well look. When I evaluate the function at a number 36 00:02:41,089 --> 00:02:45,037 very close to 0, the output is negative. So the limit is probably not some 37 00:02:45,037 --> 00:02:49,089 positive number but there's also inputs very close to zero that give positive 38 00:02:49,089 --> 00:02:52,517 outputs so that the limit is pulling out a negative number either. 39 00:02:52,517 --> 00:02:56,621 Limit is pulling out zero either cause none of these numbers are getting close 40 00:02:56,621 --> 00:02:57,920 to zero. So in this sense. 41 00:02:57,920 --> 00:03:02,049 This limit just doesn't exist because it's not the case that this limit is 42 00:03:02,049 --> 00:03:06,013 equal to anything in particular. If you tell me this limit is equal to l, 43 00:03:06,013 --> 00:03:09,592 I'm going to show you numbers close to zero which aren't close to l. 44 00:03:09,592 --> 00:03:12,180 Let's see another example along the same lines. 45 00:03:12,180 --> 00:03:18,713 This is a particularly confusing example because in the function f(x)x) = sine pi 46 00:03:18,713 --> 00:03:22,582 / x. The function evaluated at 1 is 0. 47 00:03:22,582 --> 00:03:28,600 That's pretty clear because that is sine of pi and sine of pi is zero. 48 00:03:28,600 --> 00:03:33,053 About the function at 0.1, I'm counting that's also equal to zero. 49 00:03:33,053 --> 00:03:37,781 The function at 0.01, that is also zero. This can be kind of confusing. 50 00:03:37,781 --> 00:03:43,262 When you take a look here, I typed in sin pi divided by 0.01 on to my calculator. 51 00:03:43,262 --> 00:03:46,620 This is calculating the function's value at 01.. 52 00:03:46,620 --> 00:03:51,944 If I ask my calculator to do this, it is not telling me the answer zero, right? 53 00:03:51,944 --> 00:03:56,853 The calculator's giving me this, admittedly, a very small number, right, E 54 00:03:56,853 --> 00:04:00,587 -11 here. But it's still not actually zero. 55 00:04:00,587 --> 00:04:03,698 So, can I convince you that this is even true? 56 00:04:03,698 --> 00:04:07,640 That the functions value at 01. actually is equal to zero. 57 00:04:13,060 --> 00:04:23,040 What is f (0.01)? Well it's the same as sine, of pi / 0.01. 58 00:04:23,040 --> 00:04:27,080 Now here I'm taking pi and I'm dividing it by 0.01. 59 00:04:28,200 --> 00:04:33,387 That's the same thing as what? That's the same thing as multiplying a 60 00:04:33,387 --> 00:04:36,619 100. I'm dividing by a hundredth, that's the 61 00:04:36,619 --> 00:04:41,881 same as multiplying by a 100. So this function at.01 is sine of 100 pi. 62 00:04:41,881 --> 00:04:46,317 What's sine of 100 pi? Well, think back to what the graph of 63 00:04:46,317 --> 00:04:52,732 sine looks like. Here's a graph of sine, zero to two pi. 64 00:04:52,732 --> 00:04:58,452 If I do it again. Here it is at four pi, and I drew it 65 00:04:58,452 --> 00:05:02,202 again. Here it is at six pi, and I'm going to 66 00:05:02,202 --> 00:05:07,050 keep on going. And eventually, I'm going to get to 100 67 00:05:07,050 --> 00:05:10,891 pi. And at that point, sign really is going 68 00:05:10,891 --> 00:05:15,921 to be equal to zero. Calculators are great, they're also 69 00:05:15,921 --> 00:05:20,312 terrible. This calculator can't really calculate 70 00:05:20,312 --> 00:05:26,257 with pi, all it can do is calculate with some approximation to pi. 71 00:05:26,257 --> 00:05:26,440 We can use our human mind to evaluate this function exactly. 72 00:05:26,440 --> 00:05:33,356 In light of this evidence, you might be tricked into believing that the limit of 73 00:05:33,356 --> 00:05:40,619 f(x) as x approaches 0 is equal to 0. After all these points are approaching 74 00:05:40,619 --> 00:05:45,634 zero, and function evaluating each of these points is zero. 75 00:05:45,634 --> 00:05:52,028 So maybe that means that this is true. So it looks like the limit is equal to 76 00:05:52,028 --> 00:05:55,058 zero. But, what happens if I look at some other 77 00:05:55,058 --> 00:05:58,107 points? We'll take a look at this example. 78 00:05:58,107 --> 00:06:02,446 Here's the same function, f of x equals sign of pi over x. 79 00:06:02,446 --> 00:06:08,154 This function, if I evaluate it at 75. is this, maybe a little bit mysterious 80 00:06:08,154 --> 00:06:13,634 number, negative 866. and so forth. If I evaluate this function at 075. you 81 00:06:13,634 --> 00:06:17,896 get the same thing. If I evaluate the function at 0075,. I 82 00:06:17,896 --> 00:06:22,235 get the same thing. At 00075. I get the same thing, .000075, 83 00:06:22,235 --> 00:06:25,660 I get the same thing. So, what's going on here. 84 00:06:26,740 --> 00:06:29,584 Well what is this number? I mean 0.8666. 85 00:06:29,584 --> 00:06:32,720 This isn't just some sort of random number. 86 00:06:32,720 --> 00:06:36,367 Right? This is in fact negative the square root 87 00:06:36,367 --> 00:06:41,805 of three over two. And it looks like this function at all of 88 00:06:41,805 --> 00:06:47,529 these points has the same value, negative the square root of three over two. 89 00:06:47,529 --> 00:06:53,253 So does that mean that the limit as X approaches zero of F of X, [SOUND] is 90 00:06:53,253 --> 00:06:57,222 equal to negative the square root of three over two? 91 00:06:57,222 --> 00:07:03,099 [SOUND] I mean again all of these values, .75.075.0075, these input values are 92 00:07:03,099 --> 00:07:09,320 approaching zero, and the functions value at all of those inputs is the same. 93 00:07:09,320 --> 00:07:13,754 So, what gives, is the limit zero, is a negative point A, which is it?. 94 00:07:13,754 --> 00:07:18,390 Okay, okay, I've been little bit too tricky in picking my input points. 95 00:07:18,390 --> 00:07:23,564 Since the same function, F of X equals Sin pie over X, and here, I am picking a 96 00:07:23,564 --> 00:07:27,729 collection of points, again approaching zero, .7, .07, .007, .0007. 97 00:07:27,729 --> 00:07:32,769 Its getting closer and closer to zero. But now, my output values are looking 98 00:07:32,769 --> 00:07:36,263 pretty random. I mean, they are not over the same, for 99 00:07:36,263 --> 00:07:39,488 instance. So this is, maybe some evidence, that, 100 00:07:39,488 --> 00:07:46,440 the limit. Of sine pi over x as x approaches zero. 101 00:07:46,440 --> 00:07:50,982 Doesn't exist. Here I've got a bunch of input points 102 00:07:50,982 --> 00:07:57,883 that are getting closer and closer to zero but my output values at least don't 103 00:07:57,883 --> 00:08:02,600 appear to be getting close to. We're not just learning. 104 00:08:02,600 --> 00:08:07,404 We're exploring. [SOUND] I encourage you to cook up you 105 00:08:07,404 --> 00:08:12,208 own examples. We've seen a couple examples now of where 106 00:08:12,208 --> 00:08:16,489 limits don't exist but can you come up with more? 107 00:08:16,489 --> 00:08:17,188 [SOUND]