WEBVTT 00:00:05.180 --> 00:00:17.326 Okay, we tried direct substitution. We get one squared minus one. One squared plus 00:00:17.326 --> 00:00:19.400 one minus two. 00:00:20.760 --> 00:00:27.494 And we get zero over zero. What is zero over zero? An indeterminate form. Okay, so 00:00:27.494 --> 00:00:28.480 how would we 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 00:00:40.882 --> 00:00:44.360 know how to solve this? 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. 00:00:59.870 --> 00:01:12.260 Okay, now because X is never one in a limit, it's only approaching one. 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 00:01:23.340 --> 00:01:35.635 L’Hôpital’s Rule for two specific indeterminate forms: zero over zero or infinity 00:01:35.635 --> 00:01:37.760 over infinity. 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 00:01:50.071 --> 00:01:53.880 limit as X approaches any a, 00:02:05.340 --> 00:02:11.580 so it turns out it's the limit. 00:02:21.160 --> 00:02:29.980 We take the derivative of the numerator and the derivative of the denominator. 00:02:35.840 --> 00:02:40.808 Separately, not the derivative of the whole function, derivative of the numerator 00:02:40.808 --> 00:02:45.040 divided by the derivative of the denominator. Then we find the limit. 00:02:48.200 --> 00:02:51.540 Okay, so this is L'Hôpital's rule. 00:02:51.790 --> 00:02:52.830 So let's go back to this problem. 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 00:03:11.376 --> 00:03:13.490 infinity over infinity. 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 00:03:25.438 --> 00:03:32.750 approaches one. What is the derivative of x squared minus one? 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 00:03:44.067 --> 00:03:54.142 this. Can we use direct substitution? We can. At least we try it. This is 00:03:54.142 --> 00:04:00.490 definitely, well, this is a rational function. 00:04:01.370 --> 00:04:07.632 If we get a number, we know it is continuous there, so we use direct substitution. 00:04:07.632 --> 00:04:12.290 We get two times one over two times one plus one: two thirds. 00:04:13.970 --> 00:04:22.772 This will help us solve lots of limits very easily compared with our old methods. 00:04:22.772 --> 00:04:29.510 Why didn’t I teach this in Unit 1? We didn’t know derivatives. 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 00:04:43.718 --> 00:04:50.550 a derivative? It is a limit of a difference quotient. 00:04:50.730 --> 00:05:00.721 Okay. We didn't really do ones like this before. But now we actually can solve 00:05:00.721 --> 00:05:01.490 these. 00:05:03.310 --> 00:05:14.306 As x approaches infinity, x cubed tends to infinity; subtracting three still tends 00:05:14.306 --> 00:05:25.435 to infinity. E to the x also tends to infinity. This is an indeterminate form where 00:05:25.435 --> 00:05:29.190 we can use L’Hôpital’s Rule. 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 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 00:05:54.002 --> 00:05:56.450 tend to infinity. 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. 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 00:06:22.623 --> 00:06:24.190 more time. 00:06:32.770 --> 00:06:39.002 Six over e to the x. As x approaches infinity, the denominator grows without bound. 00:06:39.002 --> 00:06:45.083 Where does that go? It goes to zero. A fixed number over a denominator tending to 00:06:45.083 --> 00:06:46.810 infinity tends to zero. 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? 00:06:52.649 --> 00:06:52.920 One. 00:06:55.920 --> 00:07:06.309 One. If you try direct substitution, sine of zero is zero. The denominator is zero, 00:07:06.309 --> 00:07:14.320 so the form is zero over zero. We can use L’Hôpital’s Rule here. 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 00:07:28.477 --> 00:07:35.880 direct substitution. Cosine of zero is? One. One. Beautiful. 00:07:36.100 --> 00:07:37.740 Okay. Go to the backside example. 00:07:39.260 --> 00:07:39.400 Three. 00:08:03.060 --> 00:08:04.460 Three. 00:08:19.260 --> 00:08:20.660 Okay. 00:08:20.660 --> 00:08:20.860 Okay. 00:08:27.600 --> 00:08:33.400 So if we try direct substitution, what is F of three? 00:08:35.820 --> 00:08:38.360 It's right here. This is three zero. 00:08:40.840 --> 00:08:43.780 And we plug in three here. We get zero. 00:08:46.320 --> 00:08:49.580 Zero over zero. 00:08:51.120 --> 00:08:51.820 Okay. 00:08:56.780 --> 00:08:58.800 So what are we going to use? 00:09:03.620 --> 00:09:04.380 Okay. 00:09:10.740 --> 00:09:14.420 Which is going to be F prime of X over two X. 00:09:18.800 --> 00:09:20.780 The denominator is easy. 00:09:23.120 --> 00:09:24.100 Okay. 00:09:24.620 --> 00:09:27.060 What is F prime of three? 00:09:31.780 --> 00:09:35.420 This is the slope of the tangent line. 00:09:40.560 --> 00:09:43.500 Of F at X equals three, right? 00:09:50.340 --> 00:09:53.380 So what is the slope of the tangent line? 00:09:54.440 --> 00:09:56.060 It's the slope of the line. 00:09:57.120 --> 00:09:58.720 And what is the slope of that line? 00:10:00.120 --> 00:10:01.180 Negative two. 00:10:02.540 --> 00:10:04.880 We get negative one-third. Beautiful. 00:10:08.220 --> 00:10:14.100 Is there another way to solve this without L’Hôpital’s Rule? 00:10:18.180 --> 00:10:19.400 We actually could. 00:10:22.840 --> 00:10:26.120 Okay. F of X is what kind of a function? 00:10:33.120 --> 00:10:34.760 Oh, is it a linear function? 00:10:36.020 --> 00:10:37.740 That doesn't look like a linear function. 00:10:39.180 --> 00:10:40.200 A piecewise function. 00:10:42.660 --> 00:10:44.160 Each piece is linear. 00:10:45.400 --> 00:10:45.420 Okay. 00:10:46.640 --> 00:10:49.300 If all we care about is three, all we care about is this piece. 00:10:49.640 --> 00:10:51.360 Could we come up with the equation of that line? 00:10:52.440 --> 00:10:54.900 Yeah. What is the equation of that line? 00:10:57.860 --> 00:11:01.140 We know the slope, which is? 00:11:01.480 --> 00:11:03.180 Negative two. 00:11:05.940 --> 00:11:08.060 And then, yeah, plus B. 00:11:09.220 --> 00:11:10.500 And then, where's that? Three zero? 00:11:11.680 --> 00:11:12.020 Uh-oh. 00:11:13.340 --> 00:11:17.380 Okay. You could come up with the equation and then plug that in and then factor.