1 00:00:05,180 --> 00:00:17,326 Okay, we tried direct substitution. We get one squared minus one. One squared plus 2 00:00:17,326 --> 00:00:19,400 one minus two. 3 00:00:20,760 --> 00:00:27,494 And we get zero over zero. What is zero over zero? An indeterminate form. Okay, so 4 00:00:27,494 --> 00:00:28,480 how would we 5 00:00:28,480 --> 00:00:40,882 solve this one? Oh, before we learn about L'Hôpital's rule, what's the only way we 6 00:00:40,882 --> 00:00:44,360 know how to solve this? 7 00:00:45,560 --> 00:00:59,870 Factor. Okay, we factor. X minus one, X plus one. X plus two, is that right? Yeah. 8 00:00:59,870 --> 00:01:12,260 Okay, now because X is never one in a limit, it's only approaching one. 9 00:01:12,260 --> 00:01:23,340 Now we plug it in: one plus one over one plus two. Two thirds. We can use 10 00:01:23,340 --> 00:01:35,635 L’Hôpital’s Rule for two specific indeterminate forms: zero over zero or infinity 11 00:01:35,635 --> 00:01:37,760 over infinity. 12 00:01:38,780 --> 00:01:50,071 Okay, and here is how L'Hôpital's rule works. If we get one of these two forms, the 13 00:01:50,071 --> 00:01:53,880 limit as X approaches any a, 14 00:02:05,340 --> 00:02:11,580 so it turns out it's the limit. 15 00:02:21,160 --> 00:02:29,980 We take the derivative of the numerator and the derivative of the denominator. 16 00:02:35,840 --> 00:02:40,808 Separately, not the derivative of the whole function, derivative of the numerator 17 00:02:40,808 --> 00:02:45,040 divided by the derivative of the denominator. Then we find the limit. 18 00:02:48,200 --> 00:02:51,540 Okay, so this is L'Hôpital's rule. 19 00:02:51,790 --> 00:02:52,830 So let's go back to this problem. 20 00:03:03,930 --> 00:03:11,376 We know it is an indeterminate form. You do need to check that: zero over zero or 21 00:03:11,376 --> 00:03:13,490 infinity over infinity. 22 00:03:16,710 --> 00:03:25,438 I always put L’Hôpital’s Rule above the equal sign. This is the limit as x 23 00:03:25,438 --> 00:03:32,750 approaches one. What is the derivative of x squared minus one? 24 00:03:32,750 --> 00:03:44,067 Two X. Two X. The derivative of this is two X plus one. Okay. And now we try solve 25 00:03:44,067 --> 00:03:54,142 this. Can we use direct substitution? We can. At least we try it. This is 26 00:03:54,142 --> 00:04:00,490 definitely, well, this is a rational function. 27 00:04:01,370 --> 00:04:07,632 If we get a number, we know it is continuous there, so we use direct substitution. 28 00:04:07,632 --> 00:04:12,290 We get two times one over two times one plus one: two thirds. 29 00:04:13,970 --> 00:04:22,772 This will help us solve lots of limits very easily compared with our old methods. 30 00:04:22,772 --> 00:04:29,510 Why didn’t I teach this in Unit 1? We didn’t know derivatives. 31 00:04:32,890 --> 00:04:43,718 There was no way to find a derivative before you knew what a derivative was. What is 32 00:04:43,718 --> 00:04:50,550 a derivative? It is a limit of a difference quotient. 33 00:04:50,730 --> 00:05:00,721 Okay. We didn't really do ones like this before. But now we actually can solve 34 00:05:00,721 --> 00:05:01,490 these. 35 00:05:03,310 --> 00:05:14,306 As x approaches infinity, x cubed tends to infinity; subtracting three still tends 36 00:05:14,306 --> 00:05:25,435 to infinity. E to the x also tends to infinity. This is an indeterminate form where 37 00:05:25,435 --> 00:05:29,190 we can use L’Hôpital’s Rule. 38 00:05:30,390 --> 00:05:42,340 Use L’Hôpital’s Rule. The derivative of x cubed minus three is three x squared. The 39 00:05:42,340 --> 00:05:54,002 derivative of e to the x is e to the x. Try again: both numerator and denominator 40 00:05:54,002 --> 00:05:56,450 tend to infinity. 41 00:05:56,450 --> 00:06:09,615 Now what? Do it again: L’Hôpital’s Rule. The derivative of three x squared is six x. 42 00:06:09,615 --> 00:06:22,623 The derivative of e to the x is e to the x. We still get infinity over infinity one 43 00:06:22,623 --> 00:06:24,190 more time. 44 00:06:32,770 --> 00:06:39,002 Six over e to the x. As x approaches infinity, the denominator grows without bound. 45 00:06:39,002 --> 00:06:45,083 Where does that go? It goes to zero. A fixed number over a denominator tending to 46 00:06:45,083 --> 00:06:46,810 infinity tends to zero. 47 00:06:47,020 --> 00:06:52,649 So we know the answer to this. It's one of the ones we memorized and the answer is? 48 00:06:52,649 --> 00:06:52,920 One. 49 00:06:55,920 --> 00:07:06,309 One. If you try direct substitution, sine of zero is zero. The denominator is zero, 50 00:07:06,309 --> 00:07:14,320 so the form is zero over zero. We can use L’Hôpital’s Rule here. 51 00:07:18,360 --> 00:07:28,477 Derivative of sine of X is? Cosine of X. Derivative of X is? One. Okay. Now we use 52 00:07:28,477 --> 00:07:35,880 direct substitution. Cosine of zero is? One. One. Beautiful. 53 00:07:36,100 --> 00:07:37,740 Okay. Go to the backside example. 54 00:07:39,260 --> 00:07:39,400 Three. 55 00:08:03,060 --> 00:08:04,460 Three. 56 00:08:19,260 --> 00:08:20,660 Okay. 57 00:08:20,660 --> 00:08:20,860 Okay. 58 00:08:27,600 --> 00:08:33,400 So if we try direct substitution, what is F of three? 59 00:08:35,820 --> 00:08:38,360 It's right here. This is three zero. 60 00:08:40,840 --> 00:08:43,780 And we plug in three here. We get zero. 61 00:08:46,320 --> 00:08:49,580 Zero over zero. 62 00:08:51,120 --> 00:08:51,820 Okay. 63 00:08:56,780 --> 00:08:58,800 So what are we going to use? 64 00:09:03,620 --> 00:09:04,380 Okay. 65 00:09:10,740 --> 00:09:14,420 Which is going to be F prime of X over two X. 66 00:09:18,800 --> 00:09:20,780 The denominator is easy. 67 00:09:23,120 --> 00:09:24,100 Okay. 68 00:09:24,620 --> 00:09:27,060 What is F prime of three? 69 00:09:31,780 --> 00:09:35,420 This is the slope of the tangent line. 70 00:09:40,560 --> 00:09:43,500 Of F at X equals three, right? 71 00:09:50,340 --> 00:09:53,380 So what is the slope of the tangent line? 72 00:09:54,440 --> 00:09:56,060 It's the slope of the line. 73 00:09:57,120 --> 00:09:58,720 And what is the slope of that line? 74 00:10:00,120 --> 00:10:01,180 Negative two. 75 00:10:02,540 --> 00:10:04,880 We get negative one-third. Beautiful. 76 00:10:08,220 --> 00:10:14,100 Is there another way to solve this without L’Hôpital’s Rule? 77 00:10:18,180 --> 00:10:19,400 We actually could. 78 00:10:22,840 --> 00:10:26,120 Okay. F of X is what kind of a function? 79 00:10:33,120 --> 00:10:34,760 Oh, is it a linear function? 80 00:10:36,020 --> 00:10:37,740 That doesn't look like a linear function. 81 00:10:39,180 --> 00:10:40,200 A piecewise function. 82 00:10:42,660 --> 00:10:44,160 Each piece is linear. 83 00:10:45,400 --> 00:10:45,420 Okay. 84 00:10:46,640 --> 00:10:49,300 If all we care about is three, all we care about is this piece. 85 00:10:49,640 --> 00:10:51,360 Could we come up with the equation of that line? 86 00:10:52,440 --> 00:10:54,900 Yeah. What is the equation of that line? 87 00:10:57,860 --> 00:11:01,140 We know the slope, which is? 88 00:11:01,480 --> 00:11:03,180 Negative two. 89 00:11:05,940 --> 00:11:08,060 And then, yeah, plus B. 90 00:11:09,220 --> 00:11:10,500 And then, where's that? Three zero? 91 00:11:11,680 --> 00:11:12,020 Uh-oh. 92 00:11:13,340 --> 00:11:17,380 Okay. You could come up with the equation and then plug that in and then factor.