1 00:00:00,000 --> 00:00:02,230 The following content is provided under a Creative 2 00:00:02,230 --> 00:00:02,896 Commons license. 3 00:00:02,896 --> 00:00:06,110 Your support will help MIT OpenCourseWare 4 00:00:06,110 --> 00:00:09,576 continue to offer high quality educational resources for free. 5 00:00:09,576 --> 00:00:12,540 To make a donation or to view additional materials 6 00:00:12,540 --> 00:00:19,130 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:19,130 --> 00:00:21,660 at ocw.mit.edu. 8 00:00:21,660 --> 00:00:24,755 PROFESSOR: So we're through with techniques of integration, 9 00:00:24,755 --> 00:00:26,880 which is really the most technical thing that we're 10 00:00:26,880 --> 00:00:28,040 going to be doing. 11 00:00:28,040 --> 00:00:37,080 And now we're just clearing up a few loose ends about calculus. 12 00:00:37,080 --> 00:00:40,090 And the one we're going to talk about today 13 00:00:40,090 --> 00:00:45,530 will allow us to deal with infinity. 14 00:00:45,530 --> 00:00:50,940 And it's what's known as L'Hôpital's Rule. 15 00:00:50,940 --> 00:00:55,990 Here's L'Hôpital's Rule. 16 00:00:55,990 --> 00:01:01,360 And that's what we're going to do today. 17 00:01:01,360 --> 00:01:14,990 L'Hôpital's Rule it's also known as L'Hospital's Rule. 18 00:01:14,990 --> 00:01:19,430 That's the same name, since the circumflex is what you 19 00:01:19,430 --> 00:01:25,249 put in French to omit the s. 20 00:01:25,249 --> 00:01:27,290 So it's the same thing, and it's still pronounced 21 00:01:27,290 --> 00:01:29,840 L'Hôpital, even if it's got an s in it. 22 00:01:29,840 --> 00:01:31,900 Alright, so that's the first thing 23 00:01:31,900 --> 00:01:33,650 you need to know about it. 24 00:01:33,650 --> 00:01:37,110 And what this method does is, it's 25 00:01:37,110 --> 00:01:55,410 a convenient way to calculate limits including some new ones. 26 00:01:55,410 --> 00:02:02,670 So it'll be convenient for the old ones. 27 00:02:02,670 --> 00:02:09,090 There are going to be some new ones and, as an example, 28 00:02:09,090 --> 00:02:14,642 you can calculate x ln x as x goes to infinity. 29 00:02:14,642 --> 00:02:16,850 You could, whoops, that's not a very interesting one, 30 00:02:16,850 --> 00:02:20,470 let's try x goes to 0 from the positive side. 31 00:02:20,470 --> 00:02:25,950 And you can calculate, for example, x e^(-x), 32 00:02:25,950 --> 00:02:30,190 as x goes to infinity. 33 00:02:30,190 --> 00:02:37,340 And, well, maybe I should include a few others. 34 00:02:37,340 --> 00:02:46,550 Maybe something like ln x / x as x goes to infinity. 35 00:02:46,550 --> 00:02:50,286 So these are some examples of things which, in fact, 36 00:02:50,286 --> 00:02:51,660 if you plug into your calculator, 37 00:02:51,660 --> 00:02:53,326 you can see what's happening with these. 38 00:02:53,326 --> 00:02:55,960 But if you want to understand them systematically, 39 00:02:55,960 --> 00:03:00,140 it's much better to have this tool of L'Hôpital's Rule. 40 00:03:00,140 --> 00:03:02,220 And certainly there isn't a proof 41 00:03:02,220 --> 00:03:05,310 just based on a calculation in a calculator. 42 00:03:05,310 --> 00:03:07,760 So now here's the idea. 43 00:03:07,760 --> 00:03:11,750 I'll illustrate the idea first with an example. 44 00:03:11,750 --> 00:03:13,380 And then we'll make it systematic. 45 00:03:13,380 --> 00:03:15,020 And then we're going to generalize it. 46 00:03:15,020 --> 00:03:17,960 We'll make it much more-- So when 47 00:03:17,960 --> 00:03:19,470 it includes these new limits, there 48 00:03:19,470 --> 00:03:21,000 are some little pieces of trickiness 49 00:03:21,000 --> 00:03:23,050 that you have to understand. 50 00:03:23,050 --> 00:03:27,770 So, let's just take an example that you 51 00:03:27,770 --> 00:03:31,070 could have done in the very first unit of this class. 52 00:03:31,070 --> 00:03:37,260 The limit as x goes to 1 of (x^10 - 1) / (x^2 - 1). 53 00:03:41,230 --> 00:03:45,370 So that's a limit that we could've handled. 54 00:03:45,370 --> 00:03:47,100 And the thing that's interesting, 55 00:03:47,100 --> 00:03:49,229 I mean, if you like this is in this category, 56 00:03:49,229 --> 00:03:51,270 that we mentioned at the beginning of the course, 57 00:03:51,270 --> 00:03:52,480 of interesting limits. 58 00:03:52,480 --> 00:03:54,860 What's interesting about it is that if you 59 00:03:54,860 --> 00:04:00,580 do this silly thing, which is just plug in x = 1, at x = 1 60 00:04:00,580 --> 00:04:02,480 you're going to get 0 / 0. 61 00:04:02,480 --> 00:04:12,720 And that's what we call an indeterminate form. 62 00:04:12,720 --> 00:04:15,340 It's just unclear what it is. 63 00:04:15,340 --> 00:04:18,150 From that plugging, in you just can't get it. 64 00:04:18,150 --> 00:04:21,360 Now, on the other hand, there's a trick for doing this. 65 00:04:21,360 --> 00:04:23,680 And this is the trick that we did 66 00:04:23,680 --> 00:04:25,350 at the beginning of the class. 67 00:04:25,350 --> 00:04:32,730 And the idea is I can divide in the numerator and denominator 68 00:04:32,730 --> 00:04:36,230 by x - 1. 69 00:04:36,230 --> 00:04:40,330 So this limit is unchanged, if I try 70 00:04:40,330 --> 00:04:45,000 to cancel the hidden factor x - 1 in the numerator 71 00:04:45,000 --> 00:04:46,400 and denominator. 72 00:04:46,400 --> 00:04:51,630 Now, we can actually carry out these ratios of polynomials 73 00:04:51,630 --> 00:04:54,580 and calculate them by long division in algebra. 74 00:04:54,580 --> 00:04:55,660 That's very, very long. 75 00:04:55,660 --> 00:04:57,110 We want to do this with calculus. 76 00:04:57,110 --> 00:04:58,710 And we already have. 77 00:04:58,710 --> 00:05:01,530 We already know that this ratio is what's 78 00:05:01,530 --> 00:05:03,420 called a difference quotient. 79 00:05:03,420 --> 00:05:06,200 And then in the limit, it tends to the derivative 80 00:05:06,200 --> 00:05:08,340 of this function. 81 00:05:08,340 --> 00:05:10,620 So the idea is that this is actually 82 00:05:10,620 --> 00:05:14,040 equal to, in the limit, now let's 83 00:05:14,040 --> 00:05:15,400 just study one piece of it. 84 00:05:15,400 --> 00:05:20,630 So if I have a function f(x), which is x^10 - 1, 85 00:05:20,630 --> 00:05:25,200 and the value at 1 happens to be equal to 0, 86 00:05:25,200 --> 00:05:31,280 then this expression that we have, which is in disguise, 87 00:05:31,280 --> 00:05:36,320 this is in disguise the difference quotient, 88 00:05:36,320 --> 00:05:42,870 tends to, as x goes to 1, the derivative, which is f'(1). 89 00:05:42,870 --> 00:05:43,620 That's what it is. 90 00:05:43,620 --> 00:05:45,203 So we know what the numerator goes to, 91 00:05:45,203 --> 00:05:47,650 and similarly we'll know what the denominator goes to. 92 00:05:47,650 --> 00:05:51,580 But what is that? 93 00:05:51,580 --> 00:05:58,460 Well, f'(x) = 10x^9. 94 00:05:58,460 --> 00:06:00,610 So we know what the answer is. 95 00:06:00,610 --> 00:06:03,550 In the numerator it's 10x^9. 96 00:06:03,550 --> 00:06:05,340 In the denominator, it's going to be 2x, 97 00:06:05,340 --> 00:06:08,340 that's the derivative of x^2 - 1. 98 00:06:08,340 --> 00:06:13,930 And then were going to have to evaluate that at x = 1. 99 00:06:13,930 --> 00:06:18,160 And so it's going to be 10/2, which is 5. 100 00:06:18,160 --> 00:06:19,350 So the answer is 5. 101 00:06:19,350 --> 00:06:23,570 And it's pretty easy to get from our techniques and knowledge 102 00:06:23,570 --> 00:06:27,510 of derivatives, using this rather clever algebraic trick. 103 00:06:27,510 --> 00:06:33,540 This business of dividing by x - 1. 104 00:06:33,540 --> 00:06:38,030 What I want to do now is just carry this method out 105 00:06:38,030 --> 00:06:39,420 systematically. 106 00:06:39,420 --> 00:06:44,290 And that's going to give us the approach to what's known 107 00:06:44,290 --> 00:06:48,860 as L'Hôpital's Rule, what-- my main subject for today. 108 00:06:48,860 --> 00:06:50,040 So here's the idea. 109 00:06:50,040 --> 00:06:51,830 Suppose we're considering, in general, 110 00:06:51,830 --> 00:06:58,860 a limit as x goes to some number a of f(x) / g(x). 111 00:06:58,860 --> 00:07:02,330 And suppose it's the bad case where we can't decide. 112 00:07:02,330 --> 00:07:03,850 So it's indeterminate. 113 00:07:03,850 --> 00:07:09,200 f(a) = g(a) = 0. 114 00:07:09,200 --> 00:07:11,709 So it would be 0 / 0. 115 00:07:11,709 --> 00:07:13,750 Now we're just going to do exactly the same thing 116 00:07:13,750 --> 00:07:15,520 we did over here. 117 00:07:15,520 --> 00:07:19,170 Namely, we're going to divide the numerator and denominator, 118 00:07:19,170 --> 00:07:21,010 and we're going to repeat that argument. 119 00:07:21,010 --> 00:07:25,390 So we have here f(x) / (x-a). 120 00:07:25,390 --> 00:07:30,240 And g(x), divided by x - a also. 121 00:07:30,240 --> 00:07:33,200 I haven't changed anything yet. 122 00:07:33,200 --> 00:07:38,120 And now I'm going to write it in this suggestive form. 123 00:07:38,120 --> 00:07:40,789 Namely, I'm going to take separately the limit 124 00:07:40,789 --> 00:07:42,330 in the numerator and the denominator. 125 00:07:42,330 --> 00:07:44,460 And I'm going to make one more shift. 126 00:07:44,460 --> 00:07:46,800 So I'm going to take the limit, as x goes 127 00:07:46,800 --> 00:07:51,256 to a in the numerator, but I'm going to write it as ( (f(x) - 128 00:07:51,256 --> 00:07:52,516 f(a)) / (x - a). 129 00:07:52,516 --> 00:07:54,640 So that's the way I'm going to write the numerator, 130 00:07:54,640 --> 00:07:57,580 and I've got to draw a much longer line here. 131 00:07:57,580 --> 00:07:59,720 So why am I allowed to do that? 132 00:07:59,720 --> 00:08:02,690 That's because f(a) = 0. 133 00:08:02,690 --> 00:08:04,340 So I didn't change this numerator 134 00:08:04,340 --> 00:08:12,210 of the numerator any by subtracting that. f(a) = 0. 135 00:08:12,210 --> 00:08:19,260 And I'll do the same thing to the denominator. 136 00:08:19,260 --> 00:08:22,650 Again, g(a) = 0, so this is OK. 137 00:08:22,650 --> 00:08:25,760 And lo and behold, I know what these limits are. 138 00:08:25,760 --> 00:08:27,900 This is f'(a) / g'(a). 139 00:08:34,110 --> 00:08:34,660 So that's it. 140 00:08:34,660 --> 00:08:36,864 That's the technique and this evaluates the limit. 141 00:08:36,864 --> 00:08:38,530 And it's not so difficult. The formula's 142 00:08:38,530 --> 00:08:40,310 pretty straightforward here. 143 00:08:40,310 --> 00:08:51,410 And it works, provided that g'(a) is not 0. 144 00:08:51,410 --> 00:08:52,590 Yeah, question. 145 00:08:52,590 --> 00:09:05,040 STUDENT: [INAUDIBLE] 146 00:09:05,040 --> 00:09:09,665 PROFESSOR: The question is, is there 147 00:09:09,665 --> 00:09:14,210 a more intuitive way of understanding this procedure. 148 00:09:14,210 --> 00:09:21,150 And I think the short answer is that there 149 00:09:21,150 --> 00:09:22,600 are other, similar, ways. 150 00:09:22,600 --> 00:09:25,520 I don't consider them to be more intuitive. 151 00:09:25,520 --> 00:09:26,940 I will be mentioning one of them, 152 00:09:26,940 --> 00:09:29,610 which is the idea of linearization, which goes 153 00:09:29,610 --> 00:09:33,019 back to what we did in Unit 2. 154 00:09:33,019 --> 00:09:35,310 I think it's very important to understand all of these, 155 00:09:35,310 --> 00:09:36,810 more or less, at once. 156 00:09:36,810 --> 00:09:38,993 But I wouldn't claim that any of these methods 157 00:09:38,993 --> 00:09:41,824 is a more intuitive one than the other. 158 00:09:41,824 --> 00:09:43,240 But basically what's happening is, 159 00:09:43,240 --> 00:09:46,340 we're looking at the linear approximation to f, at a. 160 00:09:46,340 --> 00:09:48,980 And the linear approximation to g at a. 161 00:09:48,980 --> 00:09:52,880 That's what underlies this. 162 00:09:52,880 --> 00:09:56,630 So now I get to formulate for you L'Hôpital's Rule at least 163 00:09:56,630 --> 00:09:59,650 in what I would call the easy version or, if you like, 164 00:09:59,650 --> 00:10:00,520 Version 1. 165 00:10:00,520 --> 00:10:10,290 So here's L'Hôpital's Rule. 166 00:10:10,290 --> 00:10:15,290 Version 1. 167 00:10:15,290 --> 00:10:18,480 It's not going to be quite the same as what we just did. 168 00:10:18,480 --> 00:10:20,670 It's going to be much, much better. 169 00:10:20,670 --> 00:10:22,856 And more useful. 170 00:10:22,856 --> 00:10:24,230 And what is going to take care of 171 00:10:24,230 --> 00:10:29,730 is this problem that the denominator is not 0. 172 00:10:29,730 --> 00:10:31,420 So now here's what we're going to do. 173 00:10:31,420 --> 00:10:35,370 We're going to say that it turns out that the limit as x goes 174 00:10:35,370 --> 00:10:41,860 to a of f(x) / g(x) is equal to the limit as x goes 175 00:10:41,860 --> 00:10:45,190 to a of f'(x) / g'(x). 176 00:10:48,210 --> 00:10:51,380 Now, that looks practically the same as what we said before. 177 00:10:51,380 --> 00:10:55,310 And I have to make sure that you understand when it works. 178 00:10:55,310 --> 00:11:03,430 So it works provided this is one of these undefined expressions. 179 00:11:03,430 --> 00:11:06,700 In other words, = g(a) = 0. 180 00:11:06,700 --> 00:11:11,320 So we have a 0 / 0 expression, indeterminate. 181 00:11:11,320 --> 00:11:15,280 And, also, we need one more assumption. 182 00:11:15,280 --> 00:11:30,360 And the right-hand side, the right-hand limit exists. 183 00:11:30,360 --> 00:11:33,360 Now, this is practically the same thing 184 00:11:33,360 --> 00:11:35,310 as what I said over here. 185 00:11:35,310 --> 00:11:40,840 Namely, I took the ratio of these functions, x ^ x^10 - 186 00:11:40,840 --> 00:11:42,540 1 and x^2 - 1. 187 00:11:42,540 --> 00:11:44,820 I took their derivatives, which is 188 00:11:44,820 --> 00:11:46,180 what I did right here, right. 189 00:11:46,180 --> 00:11:48,480 I just differentiated them and I took the ratio. 190 00:11:48,480 --> 00:11:50,750 This is way easier than the quotient rule, 191 00:11:50,750 --> 00:11:53,780 and is nothing like the quotient rule. 192 00:11:53,780 --> 00:11:56,900 Don't think quotient rule. 193 00:11:56,900 --> 00:11:58,070 Don't think quotient rule. 194 00:11:58,070 --> 00:12:00,360 So we differentiate the numerator and denominator 195 00:12:00,360 --> 00:12:02,400 separately. 196 00:12:02,400 --> 00:12:08,030 And then I take the limit as x goes to 1 and I get 5. 197 00:12:08,030 --> 00:12:09,940 So that's what I'm claiming over here. 198 00:12:09,940 --> 00:12:11,640 I take these functions, I replace them 199 00:12:11,640 --> 00:12:13,350 with this ratio of derivatives, and then 200 00:12:13,350 --> 00:12:15,650 I take the limit instead, over here. 201 00:12:15,650 --> 00:12:17,996 And it turned out that the functions got much simpler 202 00:12:17,996 --> 00:12:19,120 when I differentiated them. 203 00:12:19,120 --> 00:12:21,200 I started with this messy object and I 204 00:12:21,200 --> 00:12:25,520 got this much easier object that I could easily evaluate. 205 00:12:25,520 --> 00:12:29,830 So that's the big game that's happening here. 206 00:12:29,830 --> 00:12:33,495 It works, if this limit makes sense and this limit exists. 207 00:12:33,495 --> 00:12:38,360 Now, notice I didn't claim that g, that the denominator had 208 00:12:38,360 --> 00:12:40,260 to be nonzero. 209 00:12:40,260 --> 00:12:41,790 So that's what's going to help us 210 00:12:41,790 --> 00:12:43,430 a little bit in a few examples. 211 00:12:43,430 --> 00:12:45,487 So let me give you a couple of examples 212 00:12:45,487 --> 00:12:46,570 and then we'll go further. 213 00:12:46,570 --> 00:12:48,470 Now, this is only Version 1. 214 00:12:48,470 --> 00:12:51,340 But first we have to understand how this one works. 215 00:12:51,340 --> 00:12:56,300 So here's another example. 216 00:12:56,300 --> 00:13:02,860 Take the limit as x goes to 0, of sin(5x) / sin(2x). 217 00:13:06,400 --> 00:13:09,320 This is another kind of example of a limit 218 00:13:09,320 --> 00:13:12,222 that we discussed in the first part of the course. 219 00:13:12,222 --> 00:13:13,930 Unfortunately, now we're reviewing stuff. 220 00:13:13,930 --> 00:13:15,980 So this should reinforce what you did there. 221 00:13:15,980 --> 00:13:20,330 This will be an easier way of thinking about it. 222 00:13:20,330 --> 00:13:25,050 So by L'Hôpital's Rule, so here's the step. 223 00:13:25,050 --> 00:13:27,870 We're going to take one of these steps. 224 00:13:27,870 --> 00:13:31,560 This is the limit, as x goes to 1, of the derivatives here. 225 00:13:31,560 --> 00:13:36,710 So that's 5 cos(5x) / (2 cos(2x)). 226 00:13:43,220 --> 00:13:46,080 The limit was 1 over there, but now it's 227 00:13:46,080 --> 00:13:48,530 0. a is 0 in this case. 228 00:13:48,530 --> 00:13:51,500 This is the number a. 229 00:13:51,500 --> 00:13:54,930 Thank you. 230 00:13:54,930 --> 00:13:58,230 So the limit as x goes to 0 is the same 231 00:13:58,230 --> 00:14:01,400 as the limit of the derivatives. 232 00:14:01,400 --> 00:14:02,620 And that's easy to evaluate. 233 00:14:02,620 --> 00:14:04,910 Cosine of 0 is 1, right. 234 00:14:04,910 --> 00:14:12,341 This is equal to 5 cos(5*0)-- And that's a multiplication 235 00:14:12,341 --> 00:14:12,840 sign. 236 00:14:12,840 --> 00:14:14,660 Maybe I should just write this as 0. 237 00:14:14,660 --> 00:14:17,240 Divided by 2 cos 0. 238 00:14:17,240 --> 00:14:24,440 But you know that that's 5/2. 239 00:14:24,440 --> 00:14:27,340 So this is how L'Hopital's method works. 240 00:14:27,340 --> 00:14:33,660 It's pretty painless. 241 00:14:33,660 --> 00:14:36,090 I'm going to give you another example, which 242 00:14:36,090 --> 00:14:38,520 shows that it works a little better than the method 243 00:14:38,520 --> 00:14:45,550 that I started out with. 244 00:14:45,550 --> 00:14:50,010 Here's what happens if we consider the function (cos x - 245 00:14:50,010 --> 00:14:51,430 1) / x^2. 246 00:14:55,470 --> 00:14:57,940 That was a little harder to deal with. 247 00:14:57,940 --> 00:15:03,060 And again, this is one of these 0 / 0 things near x = 0. 248 00:15:03,060 --> 00:15:11,200 As x tends to 0, this goes to an indeterminate form here. 249 00:15:11,200 --> 00:15:13,166 Now, according to our method, this 250 00:15:13,166 --> 00:15:15,540 is equivalent to, now I'm going to use this little wiggle 251 00:15:15,540 --> 00:15:18,090 because I don't want to write limit, limit, limit, 252 00:15:18,090 --> 00:15:20,030 limit a million times. 253 00:15:20,030 --> 00:15:22,330 So I'm going to use a little wiggle here. 254 00:15:22,330 --> 00:15:26,840 So as x goes to 0, this is going to behave the same way 255 00:15:26,840 --> 00:15:30,390 as differentiating numerator and denominator. 256 00:15:30,390 --> 00:15:33,640 So again this is going to be -sin x in the numerator. 257 00:15:33,640 --> 00:15:42,110 In the denominator, it's going to be 2x. 258 00:15:42,110 --> 00:15:47,140 Now, notice that we still haven't won yet. 259 00:15:47,140 --> 00:15:51,060 Because this is still of 0 / 0 type. 260 00:15:51,060 --> 00:15:54,000 When you plug in x = 0 you still get 0. 261 00:15:54,000 --> 00:15:57,270 But that doesn't damage the method. 262 00:15:57,270 --> 00:16:00,630 That doesn't make the method fail. 263 00:16:00,630 --> 00:16:10,410 This 0 / 0, we can apply L'Hôpital's Rule a second time. 264 00:16:10,410 --> 00:16:12,380 And as x goes to 0 this is the same thing 265 00:16:12,380 --> 00:16:14,860 as, again, differentiating the numerator and denominator. 266 00:16:14,860 --> 00:16:18,600 So here I get -cos x in the numerator, 267 00:16:18,600 --> 00:16:22,480 and I get 2 in the denominator. 268 00:16:22,480 --> 00:16:25,000 Again this is way easier than differentiating 269 00:16:25,000 --> 00:16:26,260 ratios of functions. 270 00:16:26,260 --> 00:16:29,240 We're only differentiating the numerator and the denominator 271 00:16:29,240 --> 00:16:33,060 separately. 272 00:16:33,060 --> 00:16:35,960 And now this is the end. 273 00:16:35,960 --> 00:16:48,660 As x goes to 0, this is - -cos 0 / 2, which is -1/2. 274 00:16:48,660 --> 00:16:53,000 Now, the justification for this comes only 275 00:16:53,000 --> 00:16:56,130 when you win in the end and get the limit. 276 00:16:56,130 --> 00:16:58,820 Because what the theorem says is that if one of these limits 277 00:16:58,820 --> 00:17:01,255 exists, then the preceding one exists. 278 00:17:01,255 --> 00:17:03,630 And once the preceding one exists, then the one before it 279 00:17:03,630 --> 00:17:04,130 exists. 280 00:17:04,130 --> 00:17:09,480 So once we know that this one exists, that works backwards. 281 00:17:09,480 --> 00:17:11,420 It applies to the preceding limit, which then 282 00:17:11,420 --> 00:17:15,030 applies to the very first one. 283 00:17:15,030 --> 00:17:17,660 And the logical structure here is a little subtle, 284 00:17:17,660 --> 00:17:19,800 which is that if the right side exists, 285 00:17:19,800 --> 00:17:25,450 then the left side will also exist. 286 00:17:25,450 --> 00:17:26,380 Yeah, question. 287 00:17:26,380 --> 00:17:32,620 STUDENT: [INAUDIBLE] 288 00:17:32,620 --> 00:17:34,950 PROFESSOR: Why does the right-hand limit have to exist, 289 00:17:34,950 --> 00:17:37,120 isn't it just the derivative that has to exist? 290 00:17:37,120 --> 00:17:38,470 No. 291 00:17:38,470 --> 00:17:40,352 The derivative of the numerator has to exist. 292 00:17:40,352 --> 00:17:42,310 The derivative of the denominator has to exist. 293 00:17:42,310 --> 00:17:45,340 And this limit has to exist. 294 00:17:45,340 --> 00:17:47,280 What doesn't have to exist, by the way, 295 00:17:47,280 --> 00:17:50,690 I never said that f prime of a has to exist. 296 00:17:50,690 --> 00:17:53,690 In fact, it's much, much more subtle. 297 00:17:53,690 --> 00:17:55,710 I'm not claiming that f'(a) exists, 298 00:17:55,710 --> 00:17:58,060 because in order to evaluate this limit, 299 00:17:58,060 --> 00:18:01,080 f'(a) need not exist. 300 00:18:01,080 --> 00:18:05,460 What has to happen is that nearby, for x not equal to a, 301 00:18:05,460 --> 00:18:06,820 these things exist. 302 00:18:06,820 --> 00:18:09,530 And then the limit has to exist. 303 00:18:09,530 --> 00:18:12,060 So there's no requirements that the limits exist. 304 00:18:12,060 --> 00:18:14,645 In fact, that's exactly going to be the point when 305 00:18:14,645 --> 00:18:16,860 we evaluate these limits here. 306 00:18:16,860 --> 00:18:22,760 Is we don't have to evaluate it right at the end. 307 00:18:22,760 --> 00:18:26,540 STUDENT: [INAUDIBLE] 308 00:18:26,540 --> 00:18:28,710 PROFESSOR: So the question that you're asking 309 00:18:28,710 --> 00:18:31,970 is, why is this the hypothesis of the theorem? 310 00:18:31,970 --> 00:18:34,700 In other words, why does this work? 311 00:18:34,700 --> 00:18:37,370 Well, the answer is that this is a theorem that's true. 312 00:18:37,370 --> 00:18:40,050 If you drop this hypothesis, it's totally false. 313 00:18:40,050 --> 00:18:41,700 And if you don't have this hypothesis, 314 00:18:41,700 --> 00:18:44,820 you can't use the theorem and you will get the wrong answer. 315 00:18:44,820 --> 00:18:48,320 I mean, it's hard to express it any further than that. 316 00:18:48,320 --> 00:18:52,040 So look, in many cases we tell you formulas. 317 00:18:52,040 --> 00:18:54,070 And in many cases it's so obvious 318 00:18:54,070 --> 00:18:56,750 when they're true that we don't have 319 00:18:56,750 --> 00:18:59,610 to worry about what we say. 320 00:18:59,610 --> 00:19:01,980 And indeed, there's something implicit here. 321 00:19:01,980 --> 00:19:04,400 I'm saying well, you know, if I wrote this symbol down, 322 00:19:04,400 --> 00:19:06,370 it must mean that the thing exists. 323 00:19:06,370 --> 00:19:08,280 So that's a subtle point. 324 00:19:08,280 --> 00:19:10,830 But what I'm emphasizing is that you 325 00:19:10,830 --> 00:19:13,730 don't need to know in advance that this one exists. 326 00:19:13,730 --> 00:19:18,020 You do need to know in advance that that one exists. 327 00:19:18,020 --> 00:19:19,850 Essentially, yeah. 328 00:19:19,850 --> 00:19:24,620 So that's the direction that it goes. 329 00:19:24,620 --> 00:19:27,555 You can't get away with not having this exist 330 00:19:27,555 --> 00:19:37,517 and still have the statement be true. 331 00:19:37,517 --> 00:19:38,600 Alright, another question. 332 00:19:38,600 --> 00:19:39,740 Thank you. 333 00:19:39,740 --> 00:19:47,700 STUDENT: [INAUDIBLE] 334 00:19:47,700 --> 00:19:52,910 PROFESSOR: So I'm getting a little ahead of myself, 335 00:19:52,910 --> 00:19:54,610 but let me just say. 336 00:19:54,610 --> 00:19:59,246 In these situations here, when x is going to 0 and x 337 00:19:59,246 --> 00:20:00,120 is going to infinity. 338 00:20:00,120 --> 00:20:02,020 For instance, here when x goes to 0, 339 00:20:02,020 --> 00:20:06,340 the logarithm is undefined at x = 0. 340 00:20:06,340 --> 00:20:08,240 Nevertheless, this theorem applies. 341 00:20:08,240 --> 00:20:10,170 And we'll be able to use it. 342 00:20:10,170 --> 00:20:12,520 Over here, as x goes to infinity, neither of these-- 343 00:20:12,520 --> 00:20:15,540 well, actually, come to think of it, e^(-x), if you like, 344 00:20:15,540 --> 00:20:17,890 it's equal to 0 at infinity. 345 00:20:17,890 --> 00:20:21,220 If you want to say that it has a value. 346 00:20:21,220 --> 00:20:24,530 But in fact, these expressions don't necessarily 347 00:20:24,530 --> 00:20:27,340 have values, at the ends. 348 00:20:27,340 --> 00:20:33,160 And nevertheless, the theorem applies. 349 00:20:33,160 --> 00:20:34,690 I mean, it can exist. 350 00:20:34,690 --> 00:20:36,930 It's perfectly OK for it to exist. 351 00:20:36,930 --> 00:20:37,910 It's no problem. 352 00:20:37,910 --> 00:20:39,220 It just doesn't need to exist. 353 00:20:39,220 --> 00:20:45,080 It isn't forced to exist. 354 00:20:45,080 --> 00:20:50,090 So here's a calculation which we just did. 355 00:20:50,090 --> 00:20:51,460 And we evaluated this. 356 00:20:51,460 --> 00:20:57,990 Now, I want to make a comparison with the method 357 00:20:57,990 --> 00:21:06,770 of approximation. 358 00:21:06,770 --> 00:21:11,470 In the method of approximations, this Example 2, 359 00:21:11,470 --> 00:21:14,480 which was the example with the sine function, 360 00:21:14,480 --> 00:21:16,820 we would use the following property. 361 00:21:16,820 --> 00:21:19,370 We would use sin u is approximately u. 362 00:21:19,370 --> 00:21:22,120 We would use that linear approximation. 363 00:21:22,120 --> 00:21:29,980 And then what we would have here is that sin(5x) / sin(2x) is 364 00:21:29,980 --> 00:21:35,610 approximately (5x)/(2x), which is of course 5/2. 365 00:21:35,610 --> 00:21:38,050 And this is true when u is approximately 0, 366 00:21:38,050 --> 00:21:41,150 and this is true certainly as x goes to 0, 367 00:21:41,150 --> 00:21:45,880 it's going to be a valid limit. 368 00:21:45,880 --> 00:21:50,640 So that's very similar to Example 2. 369 00:21:50,640 --> 00:21:57,400 In Example 3, we managed to look at this expression (cos x - 370 00:21:57,400 --> 00:21:59,640 1) / x^2. 371 00:21:59,640 --> 00:22:02,270 And for this one, you have to remember 372 00:22:02,270 --> 00:22:07,940 the approximation near x = 0 to the cosine function. 373 00:22:07,940 --> 00:22:15,580 And that's 1 - x^2 / 2. 374 00:22:15,580 --> 00:22:18,750 So that was the approximation, the quadratic approximation 375 00:22:18,750 --> 00:22:20,250 to the cosine function. 376 00:22:20,250 --> 00:22:22,780 And now, sure enough, this simplifies. 377 00:22:22,780 --> 00:22:32,579 This becomes (-x^2 / 2) / x^2, which is -1/2. 378 00:22:32,579 --> 00:22:34,620 So we get the same answer, which is a good thing. 379 00:22:34,620 --> 00:22:36,450 Because both of these methods are valid. 380 00:22:36,450 --> 00:22:39,520 They're consistent. 381 00:22:39,520 --> 00:22:42,940 You can see that neither of them is particularly a lot longer. 382 00:22:42,940 --> 00:22:45,490 You may have trouble remembering this property. 383 00:22:45,490 --> 00:22:51,050 But in fact it's something that you can easily derive. 384 00:22:51,050 --> 00:22:54,270 And, indeed, it's related to the second derivative 385 00:22:54,270 --> 00:22:56,540 of the cosine, as is this calculation here. 386 00:22:56,540 --> 00:23:04,260 They're almost the same amount of numerical content to them. 387 00:23:04,260 --> 00:23:11,650 So now what I'd like to do is explain to you why 388 00:23:11,650 --> 00:23:14,970 L'Hôpital's Rule works better in some cases. 389 00:23:14,970 --> 00:23:19,210 And the real value that it has is 390 00:23:19,210 --> 00:23:25,550 in handling these other more exotic limits. 391 00:23:25,550 --> 00:23:33,960 So now we're going to do L'Hôpital's Rule over again. 392 00:23:33,960 --> 00:23:35,360 And I'll handle these functions. 393 00:23:35,360 --> 00:23:40,680 But I'll have to rewrite them, but we'll just do that. 394 00:23:40,680 --> 00:23:42,100 So here's the property. 395 00:23:42,100 --> 00:23:48,130 That the limit as x goes to a of f(x) / g(x) is equal 396 00:23:48,130 --> 00:23:54,730 to the limit as x goes to a of f'(x) / g'(x). 397 00:23:54,730 --> 00:23:55,980 That's the property. 398 00:23:55,980 --> 00:23:57,990 And this is what we'll always be using. 399 00:23:57,990 --> 00:23:59,490 Very convenient thing. 400 00:23:59,490 --> 00:24:11,530 And remember it was true provided that f(a) = g(a) = 0. 401 00:24:11,530 --> 00:24:23,240 And that the right-hand side exists. 402 00:24:23,240 --> 00:24:25,257 But I claim that it works better, 403 00:24:25,257 --> 00:24:26,340 and I'll get rid of these. 404 00:24:26,340 --> 00:24:29,330 But I'll write them again to show you 405 00:24:29,330 --> 00:24:30,990 that it works for these. 406 00:24:30,990 --> 00:24:43,680 So there are other cases. 407 00:24:43,680 --> 00:24:47,250 And the other cases that are allowed are this. 408 00:24:47,250 --> 00:24:50,840 First of all, as indicated by what I just erased, 409 00:24:50,840 --> 00:24:53,570 you can allow a to be equal to plus or minus infinity. 410 00:24:53,570 --> 00:24:57,680 It's also OK. 411 00:24:57,680 --> 00:25:04,370 So you can take the limit going to the far ends 412 00:25:04,370 --> 00:25:05,110 of the universe. 413 00:25:05,110 --> 00:25:06,630 Both left and right. 414 00:25:06,630 --> 00:25:09,780 And then the other thing that you can do 415 00:25:09,780 --> 00:25:19,690 is, you can allow f(a) and g(a) to be plus or minus infinity. 416 00:25:19,690 --> 00:25:22,540 Is OK. 417 00:25:22,540 --> 00:25:27,450 So now, the point is that we can handle not just the 0 / 0 case, 418 00:25:27,450 --> 00:25:33,780 but also the infinity / infinity case. 419 00:25:33,780 --> 00:25:36,710 That's a very powerful tool, and quite different 420 00:25:36,710 --> 00:25:42,050 from the other cases. 421 00:25:42,050 --> 00:25:49,290 And the third thing is that the right-hand side 422 00:25:49,290 --> 00:25:56,500 doesn't really quite have to exist, in the ordinary sense. 423 00:25:56,500 --> 00:26:00,460 Or, it could be plus or minus infinity. 424 00:26:00,460 --> 00:26:01,970 That's also OK. 425 00:26:01,970 --> 00:26:04,270 That's still information. 426 00:26:04,270 --> 00:26:10,564 So if we can see where it goes, then we're still good. 427 00:26:10,564 --> 00:26:11,980 If it goes to plus infinity, if it 428 00:26:11,980 --> 00:26:13,688 goes to 0, if it goes to a finite number, 429 00:26:13,688 --> 00:26:16,390 if it goes to minus infinity, all of that will be OK. 430 00:26:16,390 --> 00:26:19,400 It just if it oscillates wildly that we'll be lost. 431 00:26:19,400 --> 00:26:27,500 And those calculations we'll never encounter. 432 00:26:27,500 --> 00:26:29,310 So this basically handles everything 433 00:26:29,310 --> 00:26:32,050 that you could possibly hope for. 434 00:26:32,050 --> 00:26:37,050 And it's a very convenient process. 435 00:26:37,050 --> 00:26:40,510 So let me carry out a few examples. 436 00:26:40,510 --> 00:26:44,760 And, let's see, I guess the first one that I wanted to do 437 00:26:44,760 --> 00:26:47,640 was x ln x. 438 00:26:47,640 --> 00:26:49,300 So what example are we up to. 439 00:26:49,300 --> 00:26:57,150 Example 3, so Example 4 is coming up. 440 00:26:57,150 --> 00:26:59,030 Example 4, this is one of the ones 441 00:26:59,030 --> 00:27:06,430 that I wrote at the beginning of the lecture, x ln x. 442 00:27:06,430 --> 00:27:12,930 This one was on our homework problem. 443 00:27:12,930 --> 00:27:17,740 In the limits of some calculation. 444 00:27:17,740 --> 00:27:25,729 But so this one, you have to look at it first 445 00:27:25,729 --> 00:27:27,020 to think about what it's doing. 446 00:27:27,020 --> 00:27:28,950 It's an indeterminate form, but it sort of 447 00:27:28,950 --> 00:27:30,980 looks like it's the wrong type. 448 00:27:30,980 --> 00:27:33,070 So why is it in an indeterminate form. 449 00:27:33,070 --> 00:27:38,600 This one goes to 0, and this one goes to minus infinity. 450 00:27:38,600 --> 00:27:40,460 So, excuse me, this is a product. 451 00:27:40,460 --> 00:27:45,999 It's 0 times minus infinity. 452 00:27:45,999 --> 00:27:47,790 So that's an indeterminate form, because we 453 00:27:47,790 --> 00:27:49,730 don't know whether the 0 wins or the infinity this 454 00:27:49,730 --> 00:27:51,620 could keep getting smaller and smaller and smaller, 455 00:27:51,620 --> 00:27:52,890 and this could be getting bigger and bigger bigger. 456 00:27:52,890 --> 00:27:55,420 The product could be anything in between. 457 00:27:55,420 --> 00:27:57,740 We just don't know. 458 00:27:57,740 --> 00:28:04,270 So the first step is to write this as a ratio of things, 459 00:28:04,270 --> 00:28:06,840 rather than a product of things. 460 00:28:06,840 --> 00:28:08,450 And it turns out that the way to do 461 00:28:08,450 --> 00:28:11,590 that is to use the logarithm in the numerator, 462 00:28:11,590 --> 00:28:14,360 and the 1 / x in the denominator. 463 00:28:14,360 --> 00:28:18,040 So this is a choice that I'm making here. 464 00:28:18,040 --> 00:28:23,800 Now, I've just converted it to a limit of the type minus 465 00:28:23,800 --> 00:28:28,100 infinity divided by infinity. 466 00:28:28,100 --> 00:28:30,760 Because the numerator is going to minus infinity as x goes 467 00:28:30,760 --> 00:28:37,550 to 0+ and the denominator 1 / x is going to plus infinity. 468 00:28:37,550 --> 00:28:40,300 Again, there's a competition, but now it's one of the forms 469 00:28:40,300 --> 00:28:44,210 to which L'Hôpital's Rule applies. 470 00:28:44,210 --> 00:28:49,400 Now I'm just going to apply L'Hôpital's Rule. 471 00:28:49,400 --> 00:28:54,251 And what it says is that I differentiate here. 472 00:28:54,251 --> 00:28:56,500 So I just differentiate the numerator and denominator. 473 00:28:56,500 --> 00:28:58,900 Applying L'Hôpital's Rule is a breeze. 474 00:28:58,900 --> 00:29:03,710 You just differentiate, differentiate. 475 00:29:03,710 --> 00:29:06,810 And now it just simplifies and we're done. 476 00:29:06,810 --> 00:29:14,180 This is the limit as x goes to 0+ of, well, the x^2's cancel. 477 00:29:14,180 --> 00:29:20,360 This is the same as just -x. x factors cancel. 478 00:29:20,360 --> 00:29:21,820 And so that's 0. 479 00:29:21,820 --> 00:29:24,170 The answer is that it's 0. 480 00:29:24,170 --> 00:29:30,650 So x goes to 0 faster then ln n goes to minus infinity. 481 00:29:30,650 --> 00:29:36,400 This 0 was the winner. 482 00:29:36,400 --> 00:29:44,240 Something you can't necessarily predict in advance. 483 00:29:44,240 --> 00:29:49,920 So let's do the other two examples that I wrote down. 484 00:29:49,920 --> 00:29:53,230 I'm going to do them in slightly more generality, 485 00:29:53,230 --> 00:29:57,480 because they're the most fundamental rate 486 00:29:57,480 --> 00:29:59,370 properties that you're going to need 487 00:29:59,370 --> 00:30:01,220 to know for the next section. 488 00:30:01,220 --> 00:30:03,000 Which is improper integrals. 489 00:30:03,000 --> 00:30:07,390 And also they're just very important for physical math, 490 00:30:07,390 --> 00:30:10,390 and any other kind of thing, basically. 491 00:30:10,390 --> 00:30:12,340 So here, let's just do these. 492 00:30:12,340 --> 00:30:16,210 So let's see, which one do I want to do first. 493 00:30:16,210 --> 00:30:21,179 So I wrote down the limit of x e^(-x), 494 00:30:21,179 --> 00:30:22,970 but I'm going to make it even more general. 495 00:30:22,970 --> 00:30:25,830 I'm going to make it any negative power 496 00:30:25,830 --> 00:30:30,190 here, where p is some positive constant. 497 00:30:30,190 --> 00:30:35,680 Now again, this is a product of functions, not a quotient, 498 00:30:35,680 --> 00:30:37,350 a ratio, of functions. 499 00:30:37,350 --> 00:30:41,010 And so I need to rewrite it. 500 00:30:41,010 --> 00:30:50,670 I'm going to write it as x / e^(px). p 501 00:30:50,670 --> 00:30:52,080 And now I'm going to apply, well, 502 00:30:52,080 --> 00:30:58,420 so it's of this form infinity / infinity. 503 00:30:58,420 --> 00:31:01,860 And now that's the same as the limit as x goes to infinity 504 00:31:01,860 --> 00:31:04,080 of 1 / (p e^(px)). 505 00:31:07,520 --> 00:31:08,670 So where does that go? 506 00:31:08,670 --> 00:31:10,250 As x goes to infinity. 507 00:31:10,250 --> 00:31:12,710 Now we can decide. 508 00:31:12,710 --> 00:31:14,430 The 1 stays where it is. 509 00:31:14,430 --> 00:31:23,440 And this, as x goes to infinity, goes to infinity. 510 00:31:23,440 --> 00:31:27,080 So the answer is 0. 511 00:31:27,080 --> 00:31:48,184 And the conclusion is that x grows more slowly then e^(px). 512 00:31:48,184 --> 00:31:49,100 As x goes to infinity. 513 00:31:49,100 --> 00:31:50,980 Remember, p is positive here, of course. 514 00:31:50,980 --> 00:31:53,990 It's the increasing exponentials. 515 00:31:53,990 --> 00:32:03,080 Not the decreasing ones. 516 00:32:03,080 --> 00:32:08,530 Let's do a variant of this. 517 00:32:08,530 --> 00:32:10,690 I'll do it the opposite way. 518 00:32:10,690 --> 00:32:13,580 So I'm going to call this Example 5'. 519 00:32:13,580 --> 00:32:15,550 It really doesn't give us any more information, 520 00:32:15,550 --> 00:32:18,280 but it gives you just a little bit more practice. 521 00:32:18,280 --> 00:32:26,170 So suppose I look at things the other way. 522 00:32:26,170 --> 00:32:35,260 e^(px) divided by, say, x^100. 523 00:32:35,260 --> 00:32:42,010 Now, this is an infinity / infinity example, again. 524 00:32:42,010 --> 00:32:44,820 And you can work out what it's doing. 525 00:32:44,820 --> 00:32:47,897 But there are two ways of thinking about this. 526 00:32:47,897 --> 00:32:49,480 There's the slow way and the fast way. 527 00:32:49,480 --> 00:32:54,390 The slow way is to differentiate this 100 times. 528 00:32:54,390 --> 00:32:55,530 That is, right? 529 00:32:55,530 --> 00:32:58,490 Apply L'Hôpital's Rule over and over and over and over again. 530 00:32:58,490 --> 00:33:00,430 All the way. 531 00:33:00,430 --> 00:33:03,570 It's clear that you could do it, but it's kind of a nuisance. 532 00:33:03,570 --> 00:33:06,850 So there's a much cleverer trick here. 533 00:33:06,850 --> 00:33:12,650 Which is to change this to the limit, as x goes to infinity, 534 00:33:12,650 --> 00:33:19,610 of the e ^ e^(px/100) / x, to the 100th power. 535 00:33:25,850 --> 00:33:32,590 So if you do that, then we just have one L'Hôpital's Rule step 536 00:33:32,590 --> 00:33:34,540 here. 537 00:33:34,540 --> 00:33:43,170 And that one is that this is the same as, ...as x goes 538 00:33:43,170 --> 00:33:51,110 to infinity of, well it's p/100 e^(px/100) divided by 1, 539 00:33:51,110 --> 00:33:53,270 all to the 100th power. 540 00:33:55,830 --> 00:34:02,480 That's our L'Hôpital step. 541 00:34:02,480 --> 00:34:09,310 And of course, that's (infinity / 1)^100. 542 00:34:09,310 --> 00:34:10,410 Which is infinity. 543 00:34:10,410 --> 00:34:13,920 Now, again I did this in a slightly different way 544 00:34:13,920 --> 00:34:16,800 to show you that it works with infinity as well. 545 00:34:16,800 --> 00:34:18,400 So that was this other case. 546 00:34:18,400 --> 00:34:20,730 The right-hand side can exist, or it 547 00:34:20,730 --> 00:34:22,460 can be plus or minus infinity. 548 00:34:22,460 --> 00:34:25,240 And that applies to this limit. 549 00:34:25,240 --> 00:34:27,990 And therefore, to the original limit. 550 00:34:27,990 --> 00:34:35,180 And the conclusion here is that e^(px), p > 0, 551 00:34:35,180 --> 00:34:46,340 grows faster than any power of x. 552 00:34:46,340 --> 00:34:50,290 I picked x^100, but obviously it didn't matter what power I 553 00:34:50,290 --> 00:34:52,670 picked. 554 00:34:52,670 --> 00:35:02,050 The exponents beat all the powers. 555 00:35:02,050 --> 00:35:05,120 So we have one more of the ones that I gave at the beginning 556 00:35:05,120 --> 00:35:07,780 to take care of. 557 00:35:07,780 --> 00:35:11,330 And that one is the logarithm. 558 00:35:11,330 --> 00:35:15,720 And its behavior at infinity. 559 00:35:15,720 --> 00:35:18,700 So I'll do a slight variant on that one, too. 560 00:35:18,700 --> 00:35:24,070 So we have Example 6, which is ln x, 561 00:35:24,070 --> 00:35:27,780 and instead of dividing by x, I'm going to divide by x^(1/3). 562 00:35:27,780 --> 00:35:29,990 I could divide by any positive power of x, 563 00:35:29,990 --> 00:35:32,170 we'll just do this example here. 564 00:35:32,170 --> 00:35:37,080 So now this, as x goes to infinity, 565 00:35:37,080 --> 00:35:43,230 is of the form infinity / infinity. 566 00:35:43,230 --> 00:35:46,470 And so it's equivalent to what happens 567 00:35:46,470 --> 00:35:49,250 when I differentiate numerator and denominator separately. 568 00:35:49,250 --> 00:36:00,320 And that's 1 / x, and here I have 1/3 x^(-2/3). 569 00:36:00,320 --> 00:36:03,470 1 / x, and then 1/3 x^(-2/3). 570 00:36:03,470 --> 00:36:06,600 Now, when the dust settles here and you get your exponents 571 00:36:06,600 --> 00:36:10,560 right, we have an x^(-1), and this is an x x^(-2/3), 572 00:36:10,560 --> 00:36:12,500 and that's a 1/3 becomes a 3. 573 00:36:12,500 --> 00:36:19,260 So this is what it is. 574 00:36:19,260 --> 00:36:23,000 And that's equal to 3x^(-1/3). 575 00:36:26,480 --> 00:36:27,830 Which we can decide. 576 00:36:27,830 --> 00:36:30,150 It goes to 0. 577 00:36:30,150 --> 00:36:37,240 As x goes to infinity. 578 00:36:37,240 --> 00:36:50,200 And so the conclusion is that ln x grows more slowly as x goes 579 00:36:50,200 --> 00:37:08,690 to infinity, than x x^(1/3) or any positive power of x. 580 00:37:08,690 --> 00:37:15,500 So any x^p, p positive, will work. 581 00:37:15,500 --> 00:37:17,670 So log is really slow, going to infinity. 582 00:37:17,670 --> 00:37:20,550 It's very, very gradual. 583 00:37:20,550 --> 00:37:21,420 Yeah, question. 584 00:37:21,420 --> 00:37:45,970 STUDENT: [INAUDIBLE] 585 00:37:45,970 --> 00:37:48,230 PROFESSOR: The question is, how many 586 00:37:48,230 --> 00:37:50,630 hypotheses do you need here? 587 00:37:50,630 --> 00:37:57,590 So I said that, and I think what you were asking is, 588 00:37:57,590 --> 00:38:02,940 if I have this hypothesis, can I also have this hypothesis. 589 00:38:02,940 --> 00:38:04,870 That's OK. 590 00:38:04,870 --> 00:38:08,840 I can have this hypothesis combined with this one. 591 00:38:08,840 --> 00:38:11,280 I need something about f(a) and g(a). 592 00:38:11,280 --> 00:38:14,910 I can't assume nothing about f(a) and g(a). 593 00:38:14,910 --> 00:38:18,190 So in other words, I have to be faced with either an infinity / 594 00:38:18,190 --> 00:38:24,180 infinity, or a 0 / 0 situation. 595 00:38:24,180 --> 00:38:26,650 So let's see. 596 00:38:26,650 --> 00:38:35,160 The rule applies in the 0 / 0, or infinity / infinity case. 597 00:38:35,160 --> 00:38:40,910 These are the only two cases that it applies in. 598 00:38:40,910 --> 00:38:45,370 And a can be anything. 599 00:38:45,370 --> 00:38:48,880 Including infinity. 600 00:38:48,880 --> 00:38:51,230 Plus or minus infinity. 601 00:38:51,230 --> 00:38:53,390 The rule applies in these two cases. 602 00:38:53,390 --> 00:38:58,310 So in other words, this is what f(a) / g(a) is. 603 00:38:58,310 --> 00:39:00,750 Either one of these. 604 00:39:00,750 --> 00:39:02,540 And in fact, it can be plus or minus. 605 00:39:02,540 --> 00:39:06,460 STUDENT: [INAUDIBLE] 606 00:39:06,460 --> 00:39:10,290 PROFESSOR: And the right-hand side has to be something. 607 00:39:10,290 --> 00:39:21,250 It has to be either finite or plus or minus infinity. 608 00:39:21,250 --> 00:39:23,690 So you need something. 609 00:39:23,690 --> 00:39:25,680 You need a specific value of a, you 610 00:39:25,680 --> 00:39:28,070 need to decide whether it's an indeterminate form. 611 00:39:28,070 --> 00:39:30,320 And you need the right-hand limit to exist. 612 00:39:30,320 --> 00:39:33,530 It's not hard to impose this. 613 00:39:33,530 --> 00:39:35,975 Because when you look at the right-hand side, 614 00:39:35,975 --> 00:39:37,350 you'll want to be calculating it. 615 00:39:37,350 --> 00:39:38,641 So you want to know what it is. 616 00:39:38,641 --> 00:39:47,640 So you'll never have problems confirming this hypothesis. 617 00:39:47,640 --> 00:39:51,280 Alright. 618 00:39:51,280 --> 00:39:54,480 Let me give you one more example here. 619 00:39:54,480 --> 00:39:56,600 Which is just slightly trickier. 620 00:39:56,600 --> 00:40:15,130 Which involves, so here's another indeterminate form. 621 00:40:15,130 --> 00:40:16,780 That's going to be 0^0. 622 00:40:20,351 --> 00:40:22,350 So there are lots of these things where you just 623 00:40:22,350 --> 00:40:23,690 don't know what to do. 624 00:40:23,690 --> 00:40:27,730 And they come out in various different ways. 625 00:40:27,730 --> 00:40:32,620 The simplest example of this is the limit as x goes to 0 from 626 00:40:32,620 --> 00:40:34,280 above of x^x. 627 00:40:41,430 --> 00:40:45,120 In order to work out what's happening with this one, 628 00:40:45,120 --> 00:40:47,610 we have to use a trick. 629 00:40:47,610 --> 00:40:52,650 And the trick is this is a moving exponent. 630 00:40:52,650 --> 00:40:56,220 And so it's appropriate to use base e. 631 00:40:56,220 --> 00:40:59,010 This is something that we did way back in the first unit. 632 00:40:59,010 --> 00:41:06,690 So, since we have a moving exponent, 633 00:41:06,690 --> 00:41:11,830 we're going to use base e. 634 00:41:11,830 --> 00:41:13,600 That's the good base to use whenever 635 00:41:13,600 --> 00:41:15,600 you have a moving exponent. 636 00:41:15,600 --> 00:41:18,570 And so rewrite this as x^x = e^(x ln x). 637 00:41:21,530 --> 00:41:23,700 And now, in order to figure out what's happening, 638 00:41:23,700 --> 00:41:25,250 we really only have to know what's 639 00:41:25,250 --> 00:41:32,140 going on with the exponent. 640 00:41:32,140 --> 00:41:34,100 So remember, actually we already did this. 641 00:41:34,100 --> 00:41:36,100 But I'm going to do it once more for you. 642 00:41:36,100 --> 00:41:39,340 This is ln x / (1/x). 643 00:41:39,340 --> 00:41:42,090 And that's equivalent, as x goes to 0, 644 00:41:42,090 --> 00:41:50,690 to using L'Hôpital's Rule to 1/x, and this is -1/x^2, 645 00:41:50,690 --> 00:41:54,350 which is -x, which goes to 0. 646 00:41:54,350 --> 00:41:58,030 As x goes to 0. 647 00:41:58,030 --> 00:42:01,370 And so what we have here is that this one is going 648 00:42:01,370 --> 00:42:05,140 to be equivalent to, well, it's going 649 00:42:05,140 --> 00:42:07,520 to tend to what we got over here. 650 00:42:07,520 --> 00:42:10,170 It's e^0. 651 00:42:10,170 --> 00:42:13,980 That exponent is what we want. 652 00:42:13,980 --> 00:42:18,250 As x goes to 0. 653 00:42:18,250 --> 00:42:27,190 So that's the answer This limit happens to be 1. 654 00:42:27,190 --> 00:42:28,940 That's actually relatively easy to do, 655 00:42:28,940 --> 00:42:42,700 given all of the power that we have at our hands. 656 00:42:42,700 --> 00:42:49,310 Now, let me give you one more example. 657 00:42:49,310 --> 00:42:54,280 Suppose you're trying to understand the limit of sin x / 658 00:42:54,280 --> 00:42:54,780 x^2. 659 00:42:59,060 --> 00:43:06,090 If you apply L'Hôpital's Rule, as x goes to 0, 660 00:43:06,090 --> 00:43:11,660 you're going to get cos x / (2x). 661 00:43:11,660 --> 00:43:19,270 And if you apply L'Hôpital's Rule again, as x goes to 0, 662 00:43:19,270 --> 00:43:24,920 you're going to get - sin x / 2. 663 00:43:24,920 --> 00:43:35,840 And this, as x goes to 0, goes to 0. 664 00:43:35,840 --> 00:43:39,820 On the other hand, if you look at the linear approximation 665 00:43:39,820 --> 00:43:49,470 method, linear approximation says that sin x 666 00:43:49,470 --> 00:43:55,090 is approximately x near 0. 667 00:43:55,090 --> 00:43:59,560 So that should be x / x^2. 668 00:43:59,560 --> 00:44:04,900 Which is 1 / x, which goes to infinity. 669 00:44:04,900 --> 00:44:08,140 As x goes to 0, at least from one side, 670 00:44:08,140 --> 00:44:13,100 minus infinity to the other side. 671 00:44:13,100 --> 00:44:17,320 So there's something fishy going on here, right? 672 00:44:17,320 --> 00:44:19,330 So this is fishy. 673 00:44:19,330 --> 00:44:21,520 Or maybe this is fishy, I don't know. 674 00:44:21,520 --> 00:44:26,251 So, tell me what's wrong here. 675 00:44:26,251 --> 00:44:26,750 Yeah. 676 00:44:26,750 --> 00:44:38,230 STUDENT: [INAUDIBLE] PROFESSOR: OK. 677 00:44:38,230 --> 00:44:42,615 So the claim is that the second application 678 00:44:42,615 --> 00:44:51,950 of L'Hôpital's Rule, this one, is wrong. 679 00:44:51,950 --> 00:44:54,620 And that's correct. 680 00:44:54,620 --> 00:44:56,700 And this is where you have to watch out, 681 00:44:56,700 --> 00:44:58,087 with L'Hôpital's Rule. 682 00:44:58,087 --> 00:44:59,920 This is exactly where you have to watch out. 683 00:44:59,920 --> 00:45:02,300 You have to apply the test. 684 00:45:02,300 --> 00:45:03,820 Here it's an indeterminate form. 685 00:45:03,820 --> 00:45:08,260 It's 0 / 0 before I applied the rule. 686 00:45:08,260 --> 00:45:10,630 But in order to apply the rule the second time, 687 00:45:10,630 --> 00:45:12,410 it still has to be 0 / 0. 688 00:45:12,410 --> 00:45:14,170 But this one isn't. 689 00:45:14,170 --> 00:45:19,109 This one is 1 / 0. 690 00:45:19,109 --> 00:45:20,650 It's no longer an indeterminate form. 691 00:45:20,650 --> 00:45:22,895 It's actually infinite. 692 00:45:22,895 --> 00:45:25,520 Either plus or minus, depending on the sign of the denominator. 693 00:45:25,520 --> 00:45:27,970 Which is just what this answer is. 694 00:45:27,970 --> 00:45:30,550 So the linear approximation is safe. 695 00:45:30,550 --> 00:45:35,160 And we just applied L'Hôpital's Rule wrong. 696 00:45:35,160 --> 00:45:55,760 So the moral of the story here is look before you L'Hôp. 697 00:45:55,760 --> 00:45:58,260 Alright. 698 00:45:58,260 --> 00:46:09,850 Now, let me say one more thing. 699 00:46:09,850 --> 00:46:22,900 I need to pile it on just a little bit, sorry. 700 00:46:22,900 --> 00:46:36,700 So don't use it as a crutch. 701 00:46:36,700 --> 00:46:39,110 We don't want to just get ourselves so weak, 702 00:46:39,110 --> 00:46:41,190 after being in the hospital for all this time, 703 00:46:41,190 --> 00:46:55,560 that we can't use, I'm sorry. 704 00:46:55,560 --> 00:47:00,700 So remember that you shouldn't have lost your senses. 705 00:47:00,700 --> 00:47:12,210 If you have something like this, so we'll do this one here. 706 00:47:12,210 --> 00:47:15,030 Suppose you're trying to understand what this 707 00:47:15,030 --> 00:47:18,200 does as x goes to infinity. 708 00:47:18,200 --> 00:47:25,700 Now, you could apply L'Hôpital's Rule five times, or four times. 709 00:47:25,700 --> 00:47:30,040 And get the answer here. 710 00:47:30,040 --> 00:47:33,570 But really, you should realize that the main terms are sitting 711 00:47:33,570 --> 00:47:34,870 there right in front of you. 712 00:47:34,870 --> 00:47:36,453 And that there's some algebra that you 713 00:47:36,453 --> 00:47:38,490 can do to simplify this. 714 00:47:38,490 --> 00:47:44,480 Namely, it's the same as 1 + 2/x + 1/x^5. 715 00:47:48,060 --> 00:47:51,510 And then in the denominator, well, let's see. 716 00:47:51,510 --> 00:47:53,110 It's x. 717 00:47:53,110 --> 00:47:57,574 So this would be dividing by 1/x^5 in both numerator 718 00:47:57,574 --> 00:47:58,240 and denominator. 719 00:47:58,240 --> 00:48:04,410 And here you have 1/x plus 2 over, sorry I overshot. 720 00:48:04,410 --> 00:48:06,830 But that's OK. 721 00:48:06,830 --> 00:48:09,110 2/x^5 here. 722 00:48:09,110 --> 00:48:12,650 So these are the main terms, if you like. 723 00:48:12,650 --> 00:48:18,510 And it's the same as 1 / (1/x), which is the same as x, 724 00:48:18,510 --> 00:48:21,490 and it goes to infinity. 725 00:48:21,490 --> 00:48:22,930 As x goes to infinity. 726 00:48:22,930 --> 00:48:25,130 Or, if you like, much more simply, 727 00:48:25,130 --> 00:48:29,230 just x^5 / x^4 is the main term. 728 00:48:29,230 --> 00:48:30,520 Which is x. 729 00:48:30,520 --> 00:48:31,590 Which goes to infinity. 730 00:48:31,590 --> 00:48:35,350 So don't forget your basic algebra 731 00:48:35,350 --> 00:48:37,210 when you're doing this kind of stuff. 732 00:48:37,210 --> 00:48:40,640 Use these things and don't use L'Hôpital's Rule. 733 00:48:40,640 --> 00:48:42,355 OK, see you next time.