WEBVTT 00:00:00.000 --> 00:00:02.230 The following content is provided under a Creative 00:00:02.230 --> 00:00:02.896 Commons license. 00:00:02.896 --> 00:00:06.110 Your support will help MIT OpenCourseWare 00:00:06.110 --> 00:00:09.576 continue to offer high quality educational resources for free. 00:00:09.576 --> 00:00:12.540 To make a donation or to view additional materials 00:00:12.540 --> 00:00:19.130 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:19.130 --> 00:00:21.660 at ocw.mit.edu. 00:00:21.660 --> 00:00:24.755 PROFESSOR: So we're through with techniques of integration, 00:00:24.755 --> 00:00:26.880 which is really the most technical thing that we're 00:00:26.880 --> 00:00:28.040 going to be doing. 00:00:28.040 --> 00:00:37.080 And now we're just clearing up a few loose ends about calculus. 00:00:37.080 --> 00:00:40.090 And the one we're going to talk about today 00:00:40.090 --> 00:00:45.530 will allow us to deal with infinity. 00:00:45.530 --> 00:00:50.940 And it's what's known as L'Hôpital's Rule. 00:00:50.940 --> 00:00:55.990 Here's L'Hôpital's Rule. 00:00:55.990 --> 00:01:01.360 And that's what we're going to do today. 00:01:01.360 --> 00:01:14.990 L'Hôpital's Rule it's also known as L'Hospital's Rule. 00:01:14.990 --> 00:01:19.430 That's the same name, since the circumflex is what you 00:01:19.430 --> 00:01:25.249 put in French to omit the s. 00:01:25.249 --> 00:01:27.290 So it's the same thing, and it's still pronounced 00:01:27.290 --> 00:01:29.840 L'Hôpital, even if it's got an s in it. 00:01:29.840 --> 00:01:31.900 Alright, so that's the first thing 00:01:31.900 --> 00:01:33.650 you need to know about it. 00:01:33.650 --> 00:01:37.110 And what this method does is, it's 00:01:37.110 --> 00:01:55.410 a convenient way to calculate limits including some new ones. 00:01:55.410 --> 00:02:02.670 So it'll be convenient for the old ones. 00:02:02.670 --> 00:02:09.090 There are going to be some new ones and, as an example, 00:02:09.090 --> 00:02:14.642 you can calculate x ln x as x goes to infinity. 00:02:14.642 --> 00:02:16.850 You could, whoops, that's not a very interesting one, 00:02:16.850 --> 00:02:20.470 let's try x goes to 0 from the positive side. 00:02:20.470 --> 00:02:25.950 And you can calculate, for example, x e^(-x), 00:02:25.950 --> 00:02:30.190 as x goes to infinity. 00:02:30.190 --> 00:02:37.340 And, well, maybe I should include a few others. 00:02:37.340 --> 00:02:46.550 Maybe something like ln x / x as x goes to infinity. 00:02:46.550 --> 00:02:50.286 So these are some examples of things which, in fact, 00:02:50.286 --> 00:02:51.660 if you plug into your calculator, 00:02:51.660 --> 00:02:53.326 you can see what's happening with these. 00:02:53.326 --> 00:02:55.960 But if you want to understand them systematically, 00:02:55.960 --> 00:03:00.140 it's much better to have this tool of L'Hôpital's Rule. 00:03:00.140 --> 00:03:02.220 And certainly there isn't a proof 00:03:02.220 --> 00:03:05.310 just based on a calculation in a calculator. 00:03:05.310 --> 00:03:07.760 So now here's the idea. 00:03:07.760 --> 00:03:11.750 I'll illustrate the idea first with an example. 00:03:11.750 --> 00:03:13.380 And then we'll make it systematic. 00:03:13.380 --> 00:03:15.020 And then we're going to generalize it. 00:03:15.020 --> 00:03:17.960 We'll make it much more-- So when 00:03:17.960 --> 00:03:19.470 it includes these new limits, there 00:03:19.470 --> 00:03:21.000 are some little pieces of trickiness 00:03:21.000 --> 00:03:23.050 that you have to understand. 00:03:23.050 --> 00:03:27.770 So, let's just take an example that you 00:03:27.770 --> 00:03:31.070 could have done in the very first unit of this class. 00:03:31.070 --> 00:03:37.260 The limit as x goes to 1 of (x^10 - 1) / (x^2 - 1). 00:03:41.230 --> 00:03:45.370 So that's a limit that we could've handled. 00:03:45.370 --> 00:03:47.100 And the thing that's interesting, 00:03:47.100 --> 00:03:49.229 I mean, if you like this is in this category, 00:03:49.229 --> 00:03:51.270 that we mentioned at the beginning of the course, 00:03:51.270 --> 00:03:52.480 of interesting limits. 00:03:52.480 --> 00:03:54.860 What's interesting about it is that if you 00:03:54.860 --> 00:04:00.580 do this silly thing, which is just plug in x = 1, at x = 1 00:04:00.580 --> 00:04:02.480 you're going to get 0 / 0. 00:04:02.480 --> 00:04:12.720 And that's what we call an indeterminate form. 00:04:12.720 --> 00:04:15.340 It's just unclear what it is. 00:04:15.340 --> 00:04:18.150 From that plugging, in you just can't get it. 00:04:18.150 --> 00:04:21.360 Now, on the other hand, there's a trick for doing this. 00:04:21.360 --> 00:04:23.680 And this is the trick that we did 00:04:23.680 --> 00:04:25.350 at the beginning of the class. 00:04:25.350 --> 00:04:32.730 And the idea is I can divide in the numerator and denominator 00:04:32.730 --> 00:04:36.230 by x - 1. 00:04:36.230 --> 00:04:40.330 So this limit is unchanged, if I try 00:04:40.330 --> 00:04:45.000 to cancel the hidden factor x - 1 in the numerator 00:04:45.000 --> 00:04:46.400 and denominator. 00:04:46.400 --> 00:04:51.630 Now, we can actually carry out these ratios of polynomials 00:04:51.630 --> 00:04:54.580 and calculate them by long division in algebra. 00:04:54.580 --> 00:04:55.660 That's very, very long. 00:04:55.660 --> 00:04:57.110 We want to do this with calculus. 00:04:57.110 --> 00:04:58.710 And we already have. 00:04:58.710 --> 00:05:01.530 We already know that this ratio is what's 00:05:01.530 --> 00:05:03.420 called a difference quotient. 00:05:03.420 --> 00:05:06.200 And then in the limit, it tends to the derivative 00:05:06.200 --> 00:05:08.340 of this function. 00:05:08.340 --> 00:05:10.620 So the idea is that this is actually 00:05:10.620 --> 00:05:14.040 equal to, in the limit, now let's 00:05:14.040 --> 00:05:15.400 just study one piece of it. 00:05:15.400 --> 00:05:20.630 So if I have a function f(x), which is x^10 - 1, 00:05:20.630 --> 00:05:25.200 and the value at 1 happens to be equal to 0, 00:05:25.200 --> 00:05:31.280 then this expression that we have, which is in disguise, 00:05:31.280 --> 00:05:36.320 this is in disguise the difference quotient, 00:05:36.320 --> 00:05:42.870 tends to, as x goes to 1, the derivative, which is f'(1). 00:05:42.870 --> 00:05:43.620 That's what it is. 00:05:43.620 --> 00:05:45.203 So we know what the numerator goes to, 00:05:45.203 --> 00:05:47.650 and similarly we'll know what the denominator goes to. 00:05:47.650 --> 00:05:51.580 But what is that? 00:05:51.580 --> 00:05:58.460 Well, f'(x) = 10x^9. 00:05:58.460 --> 00:06:00.610 So we know what the answer is. 00:06:00.610 --> 00:06:03.550 In the numerator it's 10x^9. 00:06:03.550 --> 00:06:05.340 In the denominator, it's going to be 2x, 00:06:05.340 --> 00:06:08.340 that's the derivative of x^2 - 1. 00:06:08.340 --> 00:06:13.930 And then were going to have to evaluate that at x = 1. 00:06:13.930 --> 00:06:18.160 And so it's going to be 10/2, which is 5. 00:06:18.160 --> 00:06:19.350 So the answer is 5. 00:06:19.350 --> 00:06:23.570 And it's pretty easy to get from our techniques and knowledge 00:06:23.570 --> 00:06:27.510 of derivatives, using this rather clever algebraic trick. 00:06:27.510 --> 00:06:33.540 This business of dividing by x - 1. 00:06:33.540 --> 00:06:38.030 What I want to do now is just carry this method out 00:06:38.030 --> 00:06:39.420 systematically. 00:06:39.420 --> 00:06:44.290 And that's going to give us the approach to what's known 00:06:44.290 --> 00:06:48.860 as L'Hôpital's Rule, what-- my main subject for today. 00:06:48.860 --> 00:06:50.040 So here's the idea. 00:06:50.040 --> 00:06:51.830 Suppose we're considering, in general, 00:06:51.830 --> 00:06:58.860 a limit as x goes to some number a of f(x) / g(x). 00:06:58.860 --> 00:07:02.330 And suppose it's the bad case where we can't decide. 00:07:02.330 --> 00:07:03.850 So it's indeterminate. 00:07:03.850 --> 00:07:09.200 f(a) = g(a) = 0. 00:07:09.200 --> 00:07:11.709 So it would be 0 / 0. 00:07:11.709 --> 00:07:13.750 Now we're just going to do exactly the same thing 00:07:13.750 --> 00:07:15.520 we did over here. 00:07:15.520 --> 00:07:19.170 Namely, we're going to divide the numerator and denominator, 00:07:19.170 --> 00:07:21.010 and we're going to repeat that argument. 00:07:21.010 --> 00:07:25.390 So we have here f(x) / (x-a). 00:07:25.390 --> 00:07:30.240 And g(x), divided by x - a also. 00:07:30.240 --> 00:07:33.200 I haven't changed anything yet. 00:07:33.200 --> 00:07:38.120 And now I'm going to write it in this suggestive form. 00:07:38.120 --> 00:07:40.789 Namely, I'm going to take separately the limit 00:07:40.789 --> 00:07:42.330 in the numerator and the denominator. 00:07:42.330 --> 00:07:44.460 And I'm going to make one more shift. 00:07:44.460 --> 00:07:46.800 So I'm going to take the limit, as x goes 00:07:46.800 --> 00:07:51.256 to a in the numerator, but I'm going to write it as ( (f(x) - 00:07:51.256 --> 00:07:52.516 f(a)) / (x - a). 00:07:52.516 --> 00:07:54.640 So that's the way I'm going to write the numerator, 00:07:54.640 --> 00:07:57.580 and I've got to draw a much longer line here. 00:07:57.580 --> 00:07:59.720 So why am I allowed to do that? 00:07:59.720 --> 00:08:02.690 That's because f(a) = 0. 00:08:02.690 --> 00:08:04.340 So I didn't change this numerator 00:08:04.340 --> 00:08:12.210 of the numerator any by subtracting that. f(a) = 0. 00:08:12.210 --> 00:08:19.260 And I'll do the same thing to the denominator. 00:08:19.260 --> 00:08:22.650 Again, g(a) = 0, so this is OK. 00:08:22.650 --> 00:08:25.760 And lo and behold, I know what these limits are. 00:08:25.760 --> 00:08:27.900 This is f'(a) / g'(a). 00:08:34.110 --> 00:08:34.660 So that's it. 00:08:34.660 --> 00:08:36.864 That's the technique and this evaluates the limit. 00:08:36.864 --> 00:08:38.530 And it's not so difficult. The formula's 00:08:38.530 --> 00:08:40.310 pretty straightforward here. 00:08:40.310 --> 00:08:51.410 And it works, provided that g'(a) is not 0. 00:08:51.410 --> 00:08:52.590 Yeah, question. 00:08:52.590 --> 00:09:05.040 STUDENT: [INAUDIBLE] 00:09:05.040 --> 00:09:09.665 PROFESSOR: The question is, is there 00:09:09.665 --> 00:09:14.210 a more intuitive way of understanding this procedure. 00:09:14.210 --> 00:09:21.150 And I think the short answer is that there 00:09:21.150 --> 00:09:22.600 are other, similar, ways. 00:09:22.600 --> 00:09:25.520 I don't consider them to be more intuitive. 00:09:25.520 --> 00:09:26.940 I will be mentioning one of them, 00:09:26.940 --> 00:09:29.610 which is the idea of linearization, which goes 00:09:29.610 --> 00:09:33.019 back to what we did in Unit 2. 00:09:33.019 --> 00:09:35.310 I think it's very important to understand all of these, 00:09:35.310 --> 00:09:36.810 more or less, at once. 00:09:36.810 --> 00:09:38.993 But I wouldn't claim that any of these methods 00:09:38.993 --> 00:09:41.824 is a more intuitive one than the other. 00:09:41.824 --> 00:09:43.240 But basically what's happening is, 00:09:43.240 --> 00:09:46.340 we're looking at the linear approximation to f, at a. 00:09:46.340 --> 00:09:48.980 And the linear approximation to g at a. 00:09:48.980 --> 00:09:52.880 That's what underlies this. 00:09:52.880 --> 00:09:56.630 So now I get to formulate for you L'Hôpital's Rule at least 00:09:56.630 --> 00:09:59.650 in what I would call the easy version or, if you like, 00:09:59.650 --> 00:10:00.520 Version 1. 00:10:00.520 --> 00:10:10.290 So here's L'Hôpital's Rule. 00:10:10.290 --> 00:10:15.290 Version 1. 00:10:15.290 --> 00:10:18.480 It's not going to be quite the same as what we just did. 00:10:18.480 --> 00:10:20.670 It's going to be much, much better. 00:10:20.670 --> 00:10:22.856 And more useful. 00:10:22.856 --> 00:10:24.230 And what is going to take care of 00:10:24.230 --> 00:10:29.730 is this problem that the denominator is not 0. 00:10:29.730 --> 00:10:31.420 So now here's what we're going to do. 00:10:31.420 --> 00:10:35.370 We're going to say that it turns out that the limit as x goes 00:10:35.370 --> 00:10:41.860 to a of f(x) / g(x) is equal to the limit as x goes 00:10:41.860 --> 00:10:45.190 to a of f'(x) / g'(x). 00:10:48.210 --> 00:10:51.380 Now, that looks practically the same as what we said before. 00:10:51.380 --> 00:10:55.310 And I have to make sure that you understand when it works. 00:10:55.310 --> 00:11:03.430 So it works provided this is one of these undefined expressions. 00:11:03.430 --> 00:11:06.700 In other words, = g(a) = 0. 00:11:06.700 --> 00:11:11.320 So we have a 0 / 0 expression, indeterminate. 00:11:11.320 --> 00:11:15.280 And, also, we need one more assumption. 00:11:15.280 --> 00:11:30.360 And the right-hand side, the right-hand limit exists. 00:11:30.360 --> 00:11:33.360 Now, this is practically the same thing 00:11:33.360 --> 00:11:35.310 as what I said over here. 00:11:35.310 --> 00:11:40.840 Namely, I took the ratio of these functions, x ^ x^10 - 00:11:40.840 --> 00:11:42.540 1 and x^2 - 1. 00:11:42.540 --> 00:11:44.820 I took their derivatives, which is 00:11:44.820 --> 00:11:46.180 what I did right here, right. 00:11:46.180 --> 00:11:48.480 I just differentiated them and I took the ratio. 00:11:48.480 --> 00:11:50.750 This is way easier than the quotient rule, 00:11:50.750 --> 00:11:53.780 and is nothing like the quotient rule. 00:11:53.780 --> 00:11:56.900 Don't think quotient rule. 00:11:56.900 --> 00:11:58.070 Don't think quotient rule. 00:11:58.070 --> 00:12:00.360 So we differentiate the numerator and denominator 00:12:00.360 --> 00:12:02.400 separately. 00:12:02.400 --> 00:12:08.030 And then I take the limit as x goes to 1 and I get 5. 00:12:08.030 --> 00:12:09.940 So that's what I'm claiming over here. 00:12:09.940 --> 00:12:11.640 I take these functions, I replace them 00:12:11.640 --> 00:12:13.350 with this ratio of derivatives, and then 00:12:13.350 --> 00:12:15.650 I take the limit instead, over here. 00:12:15.650 --> 00:12:17.996 And it turned out that the functions got much simpler 00:12:17.996 --> 00:12:19.120 when I differentiated them. 00:12:19.120 --> 00:12:21.200 I started with this messy object and I 00:12:21.200 --> 00:12:25.520 got this much easier object that I could easily evaluate. 00:12:25.520 --> 00:12:29.830 So that's the big game that's happening here. 00:12:29.830 --> 00:12:33.495 It works, if this limit makes sense and this limit exists. 00:12:33.495 --> 00:12:38.360 Now, notice I didn't claim that g, that the denominator had 00:12:38.360 --> 00:12:40.260 to be nonzero. 00:12:40.260 --> 00:12:41.790 So that's what's going to help us 00:12:41.790 --> 00:12:43.430 a little bit in a few examples. 00:12:43.430 --> 00:12:45.487 So let me give you a couple of examples 00:12:45.487 --> 00:12:46.570 and then we'll go further. 00:12:46.570 --> 00:12:48.470 Now, this is only Version 1. 00:12:48.470 --> 00:12:51.340 But first we have to understand how this one works. 00:12:51.340 --> 00:12:56.300 So here's another example. 00:12:56.300 --> 00:13:02.860 Take the limit as x goes to 0, of sin(5x) / sin(2x). 00:13:06.400 --> 00:13:09.320 This is another kind of example of a limit 00:13:09.320 --> 00:13:12.222 that we discussed in the first part of the course. 00:13:12.222 --> 00:13:13.930 Unfortunately, now we're reviewing stuff. 00:13:13.930 --> 00:13:15.980 So this should reinforce what you did there. 00:13:15.980 --> 00:13:20.330 This will be an easier way of thinking about it. 00:13:20.330 --> 00:13:25.050 So by L'Hôpital's Rule, so here's the step. 00:13:25.050 --> 00:13:27.870 We're going to take one of these steps. 00:13:27.870 --> 00:13:31.560 This is the limit, as x goes to 1, of the derivatives here. 00:13:31.560 --> 00:13:36.710 So that's 5 cos(5x) / (2 cos(2x)). 00:13:43.220 --> 00:13:46.080 The limit was 1 over there, but now it's 00:13:46.080 --> 00:13:48.530 0. a is 0 in this case. 00:13:48.530 --> 00:13:51.500 This is the number a. 00:13:51.500 --> 00:13:54.930 Thank you. 00:13:54.930 --> 00:13:58.230 So the limit as x goes to 0 is the same 00:13:58.230 --> 00:14:01.400 as the limit of the derivatives. 00:14:01.400 --> 00:14:02.620 And that's easy to evaluate. 00:14:02.620 --> 00:14:04.910 Cosine of 0 is 1, right. 00:14:04.910 --> 00:14:12.341 This is equal to 5 cos(5*0)-- And that's a multiplication 00:14:12.341 --> 00:14:12.840 sign. 00:14:12.840 --> 00:14:14.660 Maybe I should just write this as 0. 00:14:14.660 --> 00:14:17.240 Divided by 2 cos 0. 00:14:17.240 --> 00:14:24.440 But you know that that's 5/2. 00:14:24.440 --> 00:14:27.340 So this is how L'Hopital's method works. 00:14:27.340 --> 00:14:33.660 It's pretty painless. 00:14:33.660 --> 00:14:36.090 I'm going to give you another example, which 00:14:36.090 --> 00:14:38.520 shows that it works a little better than the method 00:14:38.520 --> 00:14:45.550 that I started out with. 00:14:45.550 --> 00:14:50.010 Here's what happens if we consider the function (cos x - 00:14:50.010 --> 00:14:51.430 1) / x^2. 00:14:55.470 --> 00:14:57.940 That was a little harder to deal with. 00:14:57.940 --> 00:15:03.060 And again, this is one of these 0 / 0 things near x = 0. 00:15:03.060 --> 00:15:11.200 As x tends to 0, this goes to an indeterminate form here. 00:15:11.200 --> 00:15:13.166 Now, according to our method, this 00:15:13.166 --> 00:15:15.540 is equivalent to, now I'm going to use this little wiggle 00:15:15.540 --> 00:15:18.090 because I don't want to write limit, limit, limit, 00:15:18.090 --> 00:15:20.030 limit a million times. 00:15:20.030 --> 00:15:22.330 So I'm going to use a little wiggle here. 00:15:22.330 --> 00:15:26.840 So as x goes to 0, this is going to behave the same way 00:15:26.840 --> 00:15:30.390 as differentiating numerator and denominator. 00:15:30.390 --> 00:15:33.640 So again this is going to be -sin x in the numerator. 00:15:33.640 --> 00:15:42.110 In the denominator, it's going to be 2x. 00:15:42.110 --> 00:15:47.140 Now, notice that we still haven't won yet. 00:15:47.140 --> 00:15:51.060 Because this is still of 0 / 0 type. 00:15:51.060 --> 00:15:54.000 When you plug in x = 0 you still get 0. 00:15:54.000 --> 00:15:57.270 But that doesn't damage the method. 00:15:57.270 --> 00:16:00.630 That doesn't make the method fail. 00:16:00.630 --> 00:16:10.410 This 0 / 0, we can apply L'Hôpital's Rule a second time. 00:16:10.410 --> 00:16:12.380 And as x goes to 0 this is the same thing 00:16:12.380 --> 00:16:14.860 as, again, differentiating the numerator and denominator. 00:16:14.860 --> 00:16:18.600 So here I get -cos x in the numerator, 00:16:18.600 --> 00:16:22.480 and I get 2 in the denominator. 00:16:22.480 --> 00:16:25.000 Again this is way easier than differentiating 00:16:25.000 --> 00:16:26.260 ratios of functions. 00:16:26.260 --> 00:16:29.240 We're only differentiating the numerator and the denominator 00:16:29.240 --> 00:16:33.060 separately. 00:16:33.060 --> 00:16:35.960 And now this is the end. 00:16:35.960 --> 00:16:48.660 As x goes to 0, this is - -cos 0 / 2, which is -1/2. 00:16:48.660 --> 00:16:53.000 Now, the justification for this comes only 00:16:53.000 --> 00:16:56.130 when you win in the end and get the limit. 00:16:56.130 --> 00:16:58.820 Because what the theorem says is that if one of these limits 00:16:58.820 --> 00:17:01.255 exists, then the preceding one exists. 00:17:01.255 --> 00:17:03.630 And once the preceding one exists, then the one before it 00:17:03.630 --> 00:17:04.130 exists. 00:17:04.130 --> 00:17:09.480 So once we know that this one exists, that works backwards. 00:17:09.480 --> 00:17:11.420 It applies to the preceding limit, which then 00:17:11.420 --> 00:17:15.030 applies to the very first one. 00:17:15.030 --> 00:17:17.660 And the logical structure here is a little subtle, 00:17:17.660 --> 00:17:19.800 which is that if the right side exists, 00:17:19.800 --> 00:17:25.450 then the left side will also exist. 00:17:25.450 --> 00:17:26.380 Yeah, question. 00:17:26.380 --> 00:17:32.620 STUDENT: [INAUDIBLE] 00:17:32.620 --> 00:17:34.950 PROFESSOR: Why does the right-hand limit have to exist, 00:17:34.950 --> 00:17:37.120 isn't it just the derivative that has to exist? 00:17:37.120 --> 00:17:38.470 No. 00:17:38.470 --> 00:17:40.352 The derivative of the numerator has to exist. 00:17:40.352 --> 00:17:42.310 The derivative of the denominator has to exist. 00:17:42.310 --> 00:17:45.340 And this limit has to exist. 00:17:45.340 --> 00:17:47.280 What doesn't have to exist, by the way, 00:17:47.280 --> 00:17:50.690 I never said that f prime of a has to exist. 00:17:50.690 --> 00:17:53.690 In fact, it's much, much more subtle. 00:17:53.690 --> 00:17:55.710 I'm not claiming that f'(a) exists, 00:17:55.710 --> 00:17:58.060 because in order to evaluate this limit, 00:17:58.060 --> 00:18:01.080 f'(a) need not exist. 00:18:01.080 --> 00:18:05.460 What has to happen is that nearby, for x not equal to a, 00:18:05.460 --> 00:18:06.820 these things exist. 00:18:06.820 --> 00:18:09.530 And then the limit has to exist. 00:18:09.530 --> 00:18:12.060 So there's no requirements that the limits exist. 00:18:12.060 --> 00:18:14.645 In fact, that's exactly going to be the point when 00:18:14.645 --> 00:18:16.860 we evaluate these limits here. 00:18:16.860 --> 00:18:22.760 Is we don't have to evaluate it right at the end. 00:18:22.760 --> 00:18:26.540 STUDENT: [INAUDIBLE] 00:18:26.540 --> 00:18:28.710 PROFESSOR: So the question that you're asking 00:18:28.710 --> 00:18:31.970 is, why is this the hypothesis of the theorem? 00:18:31.970 --> 00:18:34.700 In other words, why does this work? 00:18:34.700 --> 00:18:37.370 Well, the answer is that this is a theorem that's true. 00:18:37.370 --> 00:18:40.050 If you drop this hypothesis, it's totally false. 00:18:40.050 --> 00:18:41.700 And if you don't have this hypothesis, 00:18:41.700 --> 00:18:44.820 you can't use the theorem and you will get the wrong answer. 00:18:44.820 --> 00:18:48.320 I mean, it's hard to express it any further than that. 00:18:48.320 --> 00:18:52.040 So look, in many cases we tell you formulas. 00:18:52.040 --> 00:18:54.070 And in many cases it's so obvious 00:18:54.070 --> 00:18:56.750 when they're true that we don't have 00:18:56.750 --> 00:18:59.610 to worry about what we say. 00:18:59.610 --> 00:19:01.980 And indeed, there's something implicit here. 00:19:01.980 --> 00:19:04.400 I'm saying well, you know, if I wrote this symbol down, 00:19:04.400 --> 00:19:06.370 it must mean that the thing exists. 00:19:06.370 --> 00:19:08.280 So that's a subtle point. 00:19:08.280 --> 00:19:10.830 But what I'm emphasizing is that you 00:19:10.830 --> 00:19:13.730 don't need to know in advance that this one exists. 00:19:13.730 --> 00:19:18.020 You do need to know in advance that that one exists. 00:19:18.020 --> 00:19:19.850 Essentially, yeah. 00:19:19.850 --> 00:19:24.620 So that's the direction that it goes. 00:19:24.620 --> 00:19:27.555 You can't get away with not having this exist 00:19:27.555 --> 00:19:37.517 and still have the statement be true. 00:19:37.517 --> 00:19:38.600 Alright, another question. 00:19:38.600 --> 00:19:39.740 Thank you. 00:19:39.740 --> 00:19:47.700 STUDENT: [INAUDIBLE] 00:19:47.700 --> 00:19:52.910 PROFESSOR: So I'm getting a little ahead of myself, 00:19:52.910 --> 00:19:54.610 but let me just say. 00:19:54.610 --> 00:19:59.246 In these situations here, when x is going to 0 and x 00:19:59.246 --> 00:20:00.120 is going to infinity. 00:20:00.120 --> 00:20:02.020 For instance, here when x goes to 0, 00:20:02.020 --> 00:20:06.340 the logarithm is undefined at x = 0. 00:20:06.340 --> 00:20:08.240 Nevertheless, this theorem applies. 00:20:08.240 --> 00:20:10.170 And we'll be able to use it. 00:20:10.170 --> 00:20:12.520 Over here, as x goes to infinity, neither of these-- 00:20:12.520 --> 00:20:15.540 well, actually, come to think of it, e^(-x), if you like, 00:20:15.540 --> 00:20:17.890 it's equal to 0 at infinity. 00:20:17.890 --> 00:20:21.220 If you want to say that it has a value. 00:20:21.220 --> 00:20:24.530 But in fact, these expressions don't necessarily 00:20:24.530 --> 00:20:27.340 have values, at the ends. 00:20:27.340 --> 00:20:33.160 And nevertheless, the theorem applies. 00:20:33.160 --> 00:20:34.690 I mean, it can exist. 00:20:34.690 --> 00:20:36.930 It's perfectly OK for it to exist. 00:20:36.930 --> 00:20:37.910 It's no problem. 00:20:37.910 --> 00:20:39.220 It just doesn't need to exist. 00:20:39.220 --> 00:20:45.080 It isn't forced to exist. 00:20:45.080 --> 00:20:50.090 So here's a calculation which we just did. 00:20:50.090 --> 00:20:51.460 And we evaluated this. 00:20:51.460 --> 00:20:57.990 Now, I want to make a comparison with the method 00:20:57.990 --> 00:21:06.770 of approximation. 00:21:06.770 --> 00:21:11.470 In the method of approximations, this Example 2, 00:21:11.470 --> 00:21:14.480 which was the example with the sine function, 00:21:14.480 --> 00:21:16.820 we would use the following property. 00:21:16.820 --> 00:21:19.370 We would use sin u is approximately u. 00:21:19.370 --> 00:21:22.120 We would use that linear approximation. 00:21:22.120 --> 00:21:29.980 And then what we would have here is that sin(5x) / sin(2x) is 00:21:29.980 --> 00:21:35.610 approximately (5x)/(2x), which is of course 5/2. 00:21:35.610 --> 00:21:38.050 And this is true when u is approximately 0, 00:21:38.050 --> 00:21:41.150 and this is true certainly as x goes to 0, 00:21:41.150 --> 00:21:45.880 it's going to be a valid limit. 00:21:45.880 --> 00:21:50.640 So that's very similar to Example 2. 00:21:50.640 --> 00:21:57.400 In Example 3, we managed to look at this expression (cos x - 00:21:57.400 --> 00:21:59.640 1) / x^2. 00:21:59.640 --> 00:22:02.270 And for this one, you have to remember 00:22:02.270 --> 00:22:07.940 the approximation near x = 0 to the cosine function. 00:22:07.940 --> 00:22:15.580 And that's 1 - x^2 / 2. 00:22:15.580 --> 00:22:18.750 So that was the approximation, the quadratic approximation 00:22:18.750 --> 00:22:20.250 to the cosine function. 00:22:20.250 --> 00:22:22.780 And now, sure enough, this simplifies. 00:22:22.780 --> 00:22:32.579 This becomes (-x^2 / 2) / x^2, which is -1/2. 00:22:32.579 --> 00:22:34.620 So we get the same answer, which is a good thing. 00:22:34.620 --> 00:22:36.450 Because both of these methods are valid. 00:22:36.450 --> 00:22:39.520 They're consistent. 00:22:39.520 --> 00:22:42.940 You can see that neither of them is particularly a lot longer. 00:22:42.940 --> 00:22:45.490 You may have trouble remembering this property. 00:22:45.490 --> 00:22:51.050 But in fact it's something that you can easily derive. 00:22:51.050 --> 00:22:54.270 And, indeed, it's related to the second derivative 00:22:54.270 --> 00:22:56.540 of the cosine, as is this calculation here. 00:22:56.540 --> 00:23:04.260 They're almost the same amount of numerical content to them. 00:23:04.260 --> 00:23:11.650 So now what I'd like to do is explain to you why 00:23:11.650 --> 00:23:14.970 L'Hôpital's Rule works better in some cases. 00:23:14.970 --> 00:23:19.210 And the real value that it has is 00:23:19.210 --> 00:23:25.550 in handling these other more exotic limits. 00:23:25.550 --> 00:23:33.960 So now we're going to do L'Hôpital's Rule over again. 00:23:33.960 --> 00:23:35.360 And I'll handle these functions. 00:23:35.360 --> 00:23:40.680 But I'll have to rewrite them, but we'll just do that. 00:23:40.680 --> 00:23:42.100 So here's the property. 00:23:42.100 --> 00:23:48.130 That the limit as x goes to a of f(x) / g(x) is equal 00:23:48.130 --> 00:23:54.730 to the limit as x goes to a of f'(x) / g'(x). 00:23:54.730 --> 00:23:55.980 That's the property. 00:23:55.980 --> 00:23:57.990 And this is what we'll always be using. 00:23:57.990 --> 00:23:59.490 Very convenient thing. 00:23:59.490 --> 00:24:11.530 And remember it was true provided that f(a) = g(a) = 0. 00:24:11.530 --> 00:24:23.240 And that the right-hand side exists. 00:24:23.240 --> 00:24:25.257 But I claim that it works better, 00:24:25.257 --> 00:24:26.340 and I'll get rid of these. 00:24:26.340 --> 00:24:29.330 But I'll write them again to show you 00:24:29.330 --> 00:24:30.990 that it works for these. 00:24:30.990 --> 00:24:43.680 So there are other cases. 00:24:43.680 --> 00:24:47.250 And the other cases that are allowed are this. 00:24:47.250 --> 00:24:50.840 First of all, as indicated by what I just erased, 00:24:50.840 --> 00:24:53.570 you can allow a to be equal to plus or minus infinity. 00:24:53.570 --> 00:24:57.680 It's also OK. 00:24:57.680 --> 00:25:04.370 So you can take the limit going to the far ends 00:25:04.370 --> 00:25:05.110 of the universe. 00:25:05.110 --> 00:25:06.630 Both left and right. 00:25:06.630 --> 00:25:09.780 And then the other thing that you can do 00:25:09.780 --> 00:25:19.690 is, you can allow f(a) and g(a) to be plus or minus infinity. 00:25:19.690 --> 00:25:22.540 Is OK. 00:25:22.540 --> 00:25:27.450 So now, the point is that we can handle not just the 0 / 0 case, 00:25:27.450 --> 00:25:33.780 but also the infinity / infinity case. 00:25:33.780 --> 00:25:36.710 That's a very powerful tool, and quite different 00:25:36.710 --> 00:25:42.050 from the other cases. 00:25:42.050 --> 00:25:49.290 And the third thing is that the right-hand side 00:25:49.290 --> 00:25:56.500 doesn't really quite have to exist, in the ordinary sense. 00:25:56.500 --> 00:26:00.460 Or, it could be plus or minus infinity. 00:26:00.460 --> 00:26:01.970 That's also OK. 00:26:01.970 --> 00:26:04.270 That's still information. 00:26:04.270 --> 00:26:10.564 So if we can see where it goes, then we're still good. 00:26:10.564 --> 00:26:11.980 If it goes to plus infinity, if it 00:26:11.980 --> 00:26:13.688 goes to 0, if it goes to a finite number, 00:26:13.688 --> 00:26:16.390 if it goes to minus infinity, all of that will be OK. 00:26:16.390 --> 00:26:19.400 It just if it oscillates wildly that we'll be lost. 00:26:19.400 --> 00:26:27.500 And those calculations we'll never encounter. 00:26:27.500 --> 00:26:29.310 So this basically handles everything 00:26:29.310 --> 00:26:32.050 that you could possibly hope for. 00:26:32.050 --> 00:26:37.050 And it's a very convenient process. 00:26:37.050 --> 00:26:40.510 So let me carry out a few examples. 00:26:40.510 --> 00:26:44.760 And, let's see, I guess the first one that I wanted to do 00:26:44.760 --> 00:26:47.640 was x ln x. 00:26:47.640 --> 00:26:49.300 So what example are we up to. 00:26:49.300 --> 00:26:57.150 Example 3, so Example 4 is coming up. 00:26:57.150 --> 00:26:59.030 Example 4, this is one of the ones 00:26:59.030 --> 00:27:06.430 that I wrote at the beginning of the lecture, x ln x. 00:27:06.430 --> 00:27:12.930 This one was on our homework problem. 00:27:12.930 --> 00:27:17.740 In the limits of some calculation. 00:27:17.740 --> 00:27:25.729 But so this one, you have to look at it first 00:27:25.729 --> 00:27:27.020 to think about what it's doing. 00:27:27.020 --> 00:27:28.950 It's an indeterminate form, but it sort of 00:27:28.950 --> 00:27:30.980 looks like it's the wrong type. 00:27:30.980 --> 00:27:33.070 So why is it in an indeterminate form. 00:27:33.070 --> 00:27:38.600 This one goes to 0, and this one goes to minus infinity. 00:27:38.600 --> 00:27:40.460 So, excuse me, this is a product. 00:27:40.460 --> 00:27:45.999 It's 0 times minus infinity. 00:27:45.999 --> 00:27:47.790 So that's an indeterminate form, because we 00:27:47.790 --> 00:27:49.730 don't know whether the 0 wins or the infinity this 00:27:49.730 --> 00:27:51.620 could keep getting smaller and smaller and smaller, 00:27:51.620 --> 00:27:52.890 and this could be getting bigger and bigger bigger. 00:27:52.890 --> 00:27:55.420 The product could be anything in between. 00:27:55.420 --> 00:27:57.740 We just don't know. 00:27:57.740 --> 00:28:04.270 So the first step is to write this as a ratio of things, 00:28:04.270 --> 00:28:06.840 rather than a product of things. 00:28:06.840 --> 00:28:08.450 And it turns out that the way to do 00:28:08.450 --> 00:28:11.590 that is to use the logarithm in the numerator, 00:28:11.590 --> 00:28:14.360 and the 1 / x in the denominator. 00:28:14.360 --> 00:28:18.040 So this is a choice that I'm making here. 00:28:18.040 --> 00:28:23.800 Now, I've just converted it to a limit of the type minus 00:28:23.800 --> 00:28:28.100 infinity divided by infinity. 00:28:28.100 --> 00:28:30.760 Because the numerator is going to minus infinity as x goes 00:28:30.760 --> 00:28:37.550 to 0+ and the denominator 1 / x is going to plus infinity. 00:28:37.550 --> 00:28:40.300 Again, there's a competition, but now it's one of the forms 00:28:40.300 --> 00:28:44.210 to which L'Hôpital's Rule applies. 00:28:44.210 --> 00:28:49.400 Now I'm just going to apply L'Hôpital's Rule. 00:28:49.400 --> 00:28:54.251 And what it says is that I differentiate here. 00:28:54.251 --> 00:28:56.500 So I just differentiate the numerator and denominator. 00:28:56.500 --> 00:28:58.900 Applying L'Hôpital's Rule is a breeze. 00:28:58.900 --> 00:29:03.710 You just differentiate, differentiate. 00:29:03.710 --> 00:29:06.810 And now it just simplifies and we're done. 00:29:06.810 --> 00:29:14.180 This is the limit as x goes to 0+ of, well, the x^2's cancel. 00:29:14.180 --> 00:29:20.360 This is the same as just -x. x factors cancel. 00:29:20.360 --> 00:29:21.820 And so that's 0. 00:29:21.820 --> 00:29:24.170 The answer is that it's 0. 00:29:24.170 --> 00:29:30.650 So x goes to 0 faster then ln n goes to minus infinity. 00:29:30.650 --> 00:29:36.400 This 0 was the winner. 00:29:36.400 --> 00:29:44.240 Something you can't necessarily predict in advance. 00:29:44.240 --> 00:29:49.920 So let's do the other two examples that I wrote down. 00:29:49.920 --> 00:29:53.230 I'm going to do them in slightly more generality, 00:29:53.230 --> 00:29:57.480 because they're the most fundamental rate 00:29:57.480 --> 00:29:59.370 properties that you're going to need 00:29:59.370 --> 00:30:01.220 to know for the next section. 00:30:01.220 --> 00:30:03.000 Which is improper integrals. 00:30:03.000 --> 00:30:07.390 And also they're just very important for physical math, 00:30:07.390 --> 00:30:10.390 and any other kind of thing, basically. 00:30:10.390 --> 00:30:12.340 So here, let's just do these. 00:30:12.340 --> 00:30:16.210 So let's see, which one do I want to do first. 00:30:16.210 --> 00:30:21.179 So I wrote down the limit of x e^(-x), 00:30:21.179 --> 00:30:22.970 but I'm going to make it even more general. 00:30:22.970 --> 00:30:25.830 I'm going to make it any negative power 00:30:25.830 --> 00:30:30.190 here, where p is some positive constant. 00:30:30.190 --> 00:30:35.680 Now again, this is a product of functions, not a quotient, 00:30:35.680 --> 00:30:37.350 a ratio, of functions. 00:30:37.350 --> 00:30:41.010 And so I need to rewrite it. 00:30:41.010 --> 00:30:50.670 I'm going to write it as x / e^(px). p 00:30:50.670 --> 00:30:52.080 And now I'm going to apply, well, 00:30:52.080 --> 00:30:58.420 so it's of this form infinity / infinity. 00:30:58.420 --> 00:31:01.860 And now that's the same as the limit as x goes to infinity 00:31:01.860 --> 00:31:04.080 of 1 / (p e^(px)). 00:31:07.520 --> 00:31:08.670 So where does that go? 00:31:08.670 --> 00:31:10.250 As x goes to infinity. 00:31:10.250 --> 00:31:12.710 Now we can decide. 00:31:12.710 --> 00:31:14.430 The 1 stays where it is. 00:31:14.430 --> 00:31:23.440 And this, as x goes to infinity, goes to infinity. 00:31:23.440 --> 00:31:27.080 So the answer is 0. 00:31:27.080 --> 00:31:48.184 And the conclusion is that x grows more slowly then e^(px). 00:31:48.184 --> 00:31:49.100 As x goes to infinity. 00:31:49.100 --> 00:31:50.980 Remember, p is positive here, of course. 00:31:50.980 --> 00:31:53.990 It's the increasing exponentials. 00:31:53.990 --> 00:32:03.080 Not the decreasing ones. 00:32:03.080 --> 00:32:08.530 Let's do a variant of this. 00:32:08.530 --> 00:32:10.690 I'll do it the opposite way. 00:32:10.690 --> 00:32:13.580 So I'm going to call this Example 5'. 00:32:13.580 --> 00:32:15.550 It really doesn't give us any more information, 00:32:15.550 --> 00:32:18.280 but it gives you just a little bit more practice. 00:32:18.280 --> 00:32:26.170 So suppose I look at things the other way. 00:32:26.170 --> 00:32:35.260 e^(px) divided by, say, x^100. 00:32:35.260 --> 00:32:42.010 Now, this is an infinity / infinity example, again. 00:32:42.010 --> 00:32:44.820 And you can work out what it's doing. 00:32:44.820 --> 00:32:47.897 But there are two ways of thinking about this. 00:32:47.897 --> 00:32:49.480 There's the slow way and the fast way. 00:32:49.480 --> 00:32:54.390 The slow way is to differentiate this 100 times. 00:32:54.390 --> 00:32:55.530 That is, right? 00:32:55.530 --> 00:32:58.490 Apply L'Hôpital's Rule over and over and over and over again. 00:32:58.490 --> 00:33:00.430 All the way. 00:33:00.430 --> 00:33:03.570 It's clear that you could do it, but it's kind of a nuisance. 00:33:03.570 --> 00:33:06.850 So there's a much cleverer trick here. 00:33:06.850 --> 00:33:12.650 Which is to change this to the limit, as x goes to infinity, 00:33:12.650 --> 00:33:19.610 of the e ^ e^(px/100) / x, to the 100th power. 00:33:25.850 --> 00:33:32.590 So if you do that, then we just have one L'Hôpital's Rule step 00:33:32.590 --> 00:33:34.540 here. 00:33:34.540 --> 00:33:43.170 And that one is that this is the same as, ...as x goes 00:33:43.170 --> 00:33:51.110 to infinity of, well it's p/100 e^(px/100) divided by 1, 00:33:51.110 --> 00:33:53.270 all to the 100th power. 00:33:55.830 --> 00:34:02.480 That's our L'Hôpital step. 00:34:02.480 --> 00:34:09.310 And of course, that's (infinity / 1)^100. 00:34:09.310 --> 00:34:10.410 Which is infinity. 00:34:10.410 --> 00:34:13.920 Now, again I did this in a slightly different way 00:34:13.920 --> 00:34:16.800 to show you that it works with infinity as well. 00:34:16.800 --> 00:34:18.400 So that was this other case. 00:34:18.400 --> 00:34:20.730 The right-hand side can exist, or it 00:34:20.730 --> 00:34:22.460 can be plus or minus infinity. 00:34:22.460 --> 00:34:25.240 And that applies to this limit. 00:34:25.240 --> 00:34:27.990 And therefore, to the original limit. 00:34:27.990 --> 00:34:35.180 And the conclusion here is that e^(px), p > 0, 00:34:35.180 --> 00:34:46.340 grows faster than any power of x. 00:34:46.340 --> 00:34:50.290 I picked x^100, but obviously it didn't matter what power I 00:34:50.290 --> 00:34:52.670 picked. 00:34:52.670 --> 00:35:02.050 The exponents beat all the powers. 00:35:02.050 --> 00:35:05.120 So we have one more of the ones that I gave at the beginning 00:35:05.120 --> 00:35:07.780 to take care of. 00:35:07.780 --> 00:35:11.330 And that one is the logarithm. 00:35:11.330 --> 00:35:15.720 And its behavior at infinity. 00:35:15.720 --> 00:35:18.700 So I'll do a slight variant on that one, too. 00:35:18.700 --> 00:35:24.070 So we have Example 6, which is ln x, 00:35:24.070 --> 00:35:27.780 and instead of dividing by x, I'm going to divide by x^(1/3). 00:35:27.780 --> 00:35:29.990 I could divide by any positive power of x, 00:35:29.990 --> 00:35:32.170 we'll just do this example here. 00:35:32.170 --> 00:35:37.080 So now this, as x goes to infinity, 00:35:37.080 --> 00:35:43.230 is of the form infinity / infinity. 00:35:43.230 --> 00:35:46.470 And so it's equivalent to what happens 00:35:46.470 --> 00:35:49.250 when I differentiate numerator and denominator separately. 00:35:49.250 --> 00:36:00.320 And that's 1 / x, and here I have 1/3 x^(-2/3). 00:36:00.320 --> 00:36:03.470 1 / x, and then 1/3 x^(-2/3). 00:36:03.470 --> 00:36:06.600 Now, when the dust settles here and you get your exponents 00:36:06.600 --> 00:36:10.560 right, we have an x^(-1), and this is an x x^(-2/3), 00:36:10.560 --> 00:36:12.500 and that's a 1/3 becomes a 3. 00:36:12.500 --> 00:36:19.260 So this is what it is. 00:36:19.260 --> 00:36:23.000 And that's equal to 3x^(-1/3). 00:36:26.480 --> 00:36:27.830 Which we can decide. 00:36:27.830 --> 00:36:30.150 It goes to 0. 00:36:30.150 --> 00:36:37.240 As x goes to infinity. 00:36:37.240 --> 00:36:50.200 And so the conclusion is that ln x grows more slowly as x goes 00:36:50.200 --> 00:37:08.690 to infinity, than x x^(1/3) or any positive power of x. 00:37:08.690 --> 00:37:15.500 So any x^p, p positive, will work. 00:37:15.500 --> 00:37:17.670 So log is really slow, going to infinity. 00:37:17.670 --> 00:37:20.550 It's very, very gradual. 00:37:20.550 --> 00:37:21.420 Yeah, question. 00:37:21.420 --> 00:37:45.970 STUDENT: [INAUDIBLE] 00:37:45.970 --> 00:37:48.230 PROFESSOR: The question is, how many 00:37:48.230 --> 00:37:50.630 hypotheses do you need here? 00:37:50.630 --> 00:37:57.590 So I said that, and I think what you were asking is, 00:37:57.590 --> 00:38:02.940 if I have this hypothesis, can I also have this hypothesis. 00:38:02.940 --> 00:38:04.870 That's OK. 00:38:04.870 --> 00:38:08.840 I can have this hypothesis combined with this one. 00:38:08.840 --> 00:38:11.280 I need something about f(a) and g(a). 00:38:11.280 --> 00:38:14.910 I can't assume nothing about f(a) and g(a). 00:38:14.910 --> 00:38:18.190 So in other words, I have to be faced with either an infinity / 00:38:18.190 --> 00:38:24.180 infinity, or a 0 / 0 situation. 00:38:24.180 --> 00:38:26.650 So let's see. 00:38:26.650 --> 00:38:35.160 The rule applies in the 0 / 0, or infinity / infinity case. 00:38:35.160 --> 00:38:40.910 These are the only two cases that it applies in. 00:38:40.910 --> 00:38:45.370 And a can be anything. 00:38:45.370 --> 00:38:48.880 Including infinity. 00:38:48.880 --> 00:38:51.230 Plus or minus infinity. 00:38:51.230 --> 00:38:53.390 The rule applies in these two cases. 00:38:53.390 --> 00:38:58.310 So in other words, this is what f(a) / g(a) is. 00:38:58.310 --> 00:39:00.750 Either one of these. 00:39:00.750 --> 00:39:02.540 And in fact, it can be plus or minus. 00:39:02.540 --> 00:39:06.460 STUDENT: [INAUDIBLE] 00:39:06.460 --> 00:39:10.290 PROFESSOR: And the right-hand side has to be something. 00:39:10.290 --> 00:39:21.250 It has to be either finite or plus or minus infinity. 00:39:21.250 --> 00:39:23.690 So you need something. 00:39:23.690 --> 00:39:25.680 You need a specific value of a, you 00:39:25.680 --> 00:39:28.070 need to decide whether it's an indeterminate form. 00:39:28.070 --> 00:39:30.320 And you need the right-hand limit to exist. 00:39:30.320 --> 00:39:33.530 It's not hard to impose this. 00:39:33.530 --> 00:39:35.975 Because when you look at the right-hand side, 00:39:35.975 --> 00:39:37.350 you'll want to be calculating it. 00:39:37.350 --> 00:39:38.641 So you want to know what it is. 00:39:38.641 --> 00:39:47.640 So you'll never have problems confirming this hypothesis. 00:39:47.640 --> 00:39:51.280 Alright. 00:39:51.280 --> 00:39:54.480 Let me give you one more example here. 00:39:54.480 --> 00:39:56.600 Which is just slightly trickier. 00:39:56.600 --> 00:40:15.130 Which involves, so here's another indeterminate form. 00:40:15.130 --> 00:40:16.780 That's going to be 0^0. 00:40:20.351 --> 00:40:22.350 So there are lots of these things where you just 00:40:22.350 --> 00:40:23.690 don't know what to do. 00:40:23.690 --> 00:40:27.730 And they come out in various different ways. 00:40:27.730 --> 00:40:32.620 The simplest example of this is the limit as x goes to 0 from 00:40:32.620 --> 00:40:34.280 above of x^x. 00:40:41.430 --> 00:40:45.120 In order to work out what's happening with this one, 00:40:45.120 --> 00:40:47.610 we have to use a trick. 00:40:47.610 --> 00:40:52.650 And the trick is this is a moving exponent. 00:40:52.650 --> 00:40:56.220 And so it's appropriate to use base e. 00:40:56.220 --> 00:40:59.010 This is something that we did way back in the first unit. 00:40:59.010 --> 00:41:06.690 So, since we have a moving exponent, 00:41:06.690 --> 00:41:11.830 we're going to use base e. 00:41:11.830 --> 00:41:13.600 That's the good base to use whenever 00:41:13.600 --> 00:41:15.600 you have a moving exponent. 00:41:15.600 --> 00:41:18.570 And so rewrite this as x^x = e^(x ln x). 00:41:21.530 --> 00:41:23.700 And now, in order to figure out what's happening, 00:41:23.700 --> 00:41:25.250 we really only have to know what's 00:41:25.250 --> 00:41:32.140 going on with the exponent. 00:41:32.140 --> 00:41:34.100 So remember, actually we already did this. 00:41:34.100 --> 00:41:36.100 But I'm going to do it once more for you. 00:41:36.100 --> 00:41:39.340 This is ln x / (1/x). 00:41:39.340 --> 00:41:42.090 And that's equivalent, as x goes to 0, 00:41:42.090 --> 00:41:50.690 to using L'Hôpital's Rule to 1/x, and this is -1/x^2, 00:41:50.690 --> 00:41:54.350 which is -x, which goes to 0. 00:41:54.350 --> 00:41:58.030 As x goes to 0. 00:41:58.030 --> 00:42:01.370 And so what we have here is that this one is going 00:42:01.370 --> 00:42:05.140 to be equivalent to, well, it's going 00:42:05.140 --> 00:42:07.520 to tend to what we got over here. 00:42:07.520 --> 00:42:10.170 It's e^0. 00:42:10.170 --> 00:42:13.980 That exponent is what we want. 00:42:13.980 --> 00:42:18.250 As x goes to 0. 00:42:18.250 --> 00:42:27.190 So that's the answer This limit happens to be 1. 00:42:27.190 --> 00:42:28.940 That's actually relatively easy to do, 00:42:28.940 --> 00:42:42.700 given all of the power that we have at our hands. 00:42:42.700 --> 00:42:49.310 Now, let me give you one more example. 00:42:49.310 --> 00:42:54.280 Suppose you're trying to understand the limit of sin x / 00:42:54.280 --> 00:42:54.780 x^2. 00:42:59.060 --> 00:43:06.090 If you apply L'Hôpital's Rule, as x goes to 0, 00:43:06.090 --> 00:43:11.660 you're going to get cos x / (2x). 00:43:11.660 --> 00:43:19.270 And if you apply L'Hôpital's Rule again, as x goes to 0, 00:43:19.270 --> 00:43:24.920 you're going to get - sin x / 2. 00:43:24.920 --> 00:43:35.840 And this, as x goes to 0, goes to 0. 00:43:35.840 --> 00:43:39.820 On the other hand, if you look at the linear approximation 00:43:39.820 --> 00:43:49.470 method, linear approximation says that sin x 00:43:49.470 --> 00:43:55.090 is approximately x near 0. 00:43:55.090 --> 00:43:59.560 So that should be x / x^2. 00:43:59.560 --> 00:44:04.900 Which is 1 / x, which goes to infinity. 00:44:04.900 --> 00:44:08.140 As x goes to 0, at least from one side, 00:44:08.140 --> 00:44:13.100 minus infinity to the other side. 00:44:13.100 --> 00:44:17.320 So there's something fishy going on here, right? 00:44:17.320 --> 00:44:19.330 So this is fishy. 00:44:19.330 --> 00:44:21.520 Or maybe this is fishy, I don't know. 00:44:21.520 --> 00:44:26.251 So, tell me what's wrong here. 00:44:26.251 --> 00:44:26.750 Yeah. 00:44:26.750 --> 00:44:38.230 STUDENT: [INAUDIBLE] PROFESSOR: OK. 00:44:38.230 --> 00:44:42.615 So the claim is that the second application 00:44:42.615 --> 00:44:51.950 of L'Hôpital's Rule, this one, is wrong. 00:44:51.950 --> 00:44:54.620 And that's correct. 00:44:54.620 --> 00:44:56.700 And this is where you have to watch out, 00:44:56.700 --> 00:44:58.087 with L'Hôpital's Rule. 00:44:58.087 --> 00:44:59.920 This is exactly where you have to watch out. 00:44:59.920 --> 00:45:02.300 You have to apply the test. 00:45:02.300 --> 00:45:03.820 Here it's an indeterminate form. 00:45:03.820 --> 00:45:08.260 It's 0 / 0 before I applied the rule. 00:45:08.260 --> 00:45:10.630 But in order to apply the rule the second time, 00:45:10.630 --> 00:45:12.410 it still has to be 0 / 0. 00:45:12.410 --> 00:45:14.170 But this one isn't. 00:45:14.170 --> 00:45:19.109 This one is 1 / 0. 00:45:19.109 --> 00:45:20.650 It's no longer an indeterminate form. 00:45:20.650 --> 00:45:22.895 It's actually infinite. 00:45:22.895 --> 00:45:25.520 Either plus or minus, depending on the sign of the denominator. 00:45:25.520 --> 00:45:27.970 Which is just what this answer is. 00:45:27.970 --> 00:45:30.550 So the linear approximation is safe. 00:45:30.550 --> 00:45:35.160 And we just applied L'Hôpital's Rule wrong. 00:45:35.160 --> 00:45:55.760 So the moral of the story here is look before you L'Hôp. 00:45:55.760 --> 00:45:58.260 Alright. 00:45:58.260 --> 00:46:09.850 Now, let me say one more thing. 00:46:09.850 --> 00:46:22.900 I need to pile it on just a little bit, sorry. 00:46:22.900 --> 00:46:36.700 So don't use it as a crutch. 00:46:36.700 --> 00:46:39.110 We don't want to just get ourselves so weak, 00:46:39.110 --> 00:46:41.190 after being in the hospital for all this time, 00:46:41.190 --> 00:46:55.560 that we can't use, I'm sorry. 00:46:55.560 --> 00:47:00.700 So remember that you shouldn't have lost your senses. 00:47:00.700 --> 00:47:12.210 If you have something like this, so we'll do this one here. 00:47:12.210 --> 00:47:15.030 Suppose you're trying to understand what this 00:47:15.030 --> 00:47:18.200 does as x goes to infinity. 00:47:18.200 --> 00:47:25.700 Now, you could apply L'Hôpital's Rule five times, or four times. 00:47:25.700 --> 00:47:30.040 And get the answer here. 00:47:30.040 --> 00:47:33.570 But really, you should realize that the main terms are sitting 00:47:33.570 --> 00:47:34.870 there right in front of you. 00:47:34.870 --> 00:47:36.453 And that there's some algebra that you 00:47:36.453 --> 00:47:38.490 can do to simplify this. 00:47:38.490 --> 00:47:44.480 Namely, it's the same as 1 + 2/x + 1/x^5. 00:47:48.060 --> 00:47:51.510 And then in the denominator, well, let's see. 00:47:51.510 --> 00:47:53.110 It's x. 00:47:53.110 --> 00:47:57.574 So this would be dividing by 1/x^5 in both numerator 00:47:57.574 --> 00:47:58.240 and denominator. 00:47:58.240 --> 00:48:04.410 And here you have 1/x plus 2 over, sorry I overshot. 00:48:04.410 --> 00:48:06.830 But that's OK. 00:48:06.830 --> 00:48:09.110 2/x^5 here. 00:48:09.110 --> 00:48:12.650 So these are the main terms, if you like. 00:48:12.650 --> 00:48:18.510 And it's the same as 1 / (1/x), which is the same as x, 00:48:18.510 --> 00:48:21.490 and it goes to infinity. 00:48:21.490 --> 00:48:22.930 As x goes to infinity. 00:48:22.930 --> 00:48:25.130 Or, if you like, much more simply, 00:48:25.130 --> 00:48:29.230 just x^5 / x^4 is the main term. 00:48:29.230 --> 00:48:30.520 Which is x. 00:48:30.520 --> 00:48:31.590 Which goes to infinity. 00:48:31.590 --> 00:48:35.350 So don't forget your basic algebra 00:48:35.350 --> 00:48:37.210 when you're doing this kind of stuff. 00:48:37.210 --> 00:48:40.640 Use these things and don't use L'Hôpital's Rule. 00:48:40.640 --> 00:48:42.355 OK, see you next time.