WEBVTT 00:00:00.000 --> 00:00:07.300 PROFESSOR: Hi. 00:00:07.300 --> 00:00:08.850 Welcome back to recitation. 00:00:08.850 --> 00:00:11.200 You've been talking about computing limits of some 00:00:11.200 --> 00:00:13.710 indeterminate forms. In particular, you used 00:00:13.710 --> 00:00:17.500 l'Hopital's rule to help you out with some limits in the 00:00:17.500 --> 00:00:20.540 form 0 over 0 or infinity over infinity. 00:00:20.540 --> 00:00:22.520 So these are the two indeterminate ratios. 00:00:26.560 --> 00:00:28.190 And you also, so OK, so when you have a limit that is in 00:00:28.190 --> 00:00:30.290 the form of an indeterminate ratio, you've seen that one 00:00:30.290 --> 00:00:33.090 tool that you can use to help compute the limit is 00:00:33.090 --> 00:00:34.080 l'Hopital's rule. 00:00:34.080 --> 00:00:38.050 There are other indeterminate forms for limits as well. 00:00:38.050 --> 00:00:41.560 So you actually saw, in lecture, another one of these, 00:00:41.560 --> 00:00:45.080 which was you saw a limit of the form 0 to the 0. 00:00:45.080 --> 00:00:49.980 So that was the limit you saw, was the limit as x goes to 0 00:00:49.980 --> 00:00:52.360 from the right to x to the x. 00:00:52.360 --> 00:00:54.370 So that was going to 0 over 0. 00:00:54.370 --> 00:00:57.610 And the two competing forces are that as the base of the 00:00:57.610 --> 00:00:58.950 limit goes to 0. 00:00:58.950 --> 00:01:01.830 That wants to make the whole thing get closer to 0. 00:01:01.830 --> 00:01:04.390 And when the exponent is going to 0, that makes the whole 00:01:04.390 --> 00:01:06.080 thing want to get closer to 1. 00:01:06.080 --> 00:01:07.880 So you have those two competing forces. 00:01:07.880 --> 00:01:09.470 That's why it is an indeterminate form. 00:01:09.470 --> 00:01:12.740 When you were solving this limit, you, first thing you 00:01:12.740 --> 00:01:16.790 did was you wrote it as an exponential in base e. 00:01:16.790 --> 00:01:22.110 So you wrote x to the x as e to the xlnx. 00:01:22.110 --> 00:01:23.750 So xlnx is also an indeterminate form 00:01:23.750 --> 00:01:25.280 as x goes to 0. 00:01:25.280 --> 00:01:30.490 It's an indeterminate form of the form 0 times infinity. 00:01:30.490 --> 00:01:33.560 Here I'm writing infinity to mean either positive infinity 00:01:33.560 --> 00:01:34.910 or negative infinity. 00:01:34.910 --> 00:01:37.750 So in this case, this is an indeterminate form because 00:01:37.750 --> 00:01:40.700 when you have one factor going to 0, that makes the whole 00:01:40.700 --> 00:01:42.430 product want to get closer to 0. 00:01:42.430 --> 00:01:44.530 Whereas when you have one factor going to infinity, 00:01:44.530 --> 00:01:46.900 either positive or negative, that makes the whole product 00:01:46.900 --> 00:01:47.850 want to get big. 00:01:47.850 --> 00:01:50.140 So that's why it becomes indeterminate. 00:01:50.140 --> 00:01:54.320 And you were able to evaluate the limit xlnx, in that case, 00:01:54.320 --> 00:01:57.500 by rewriting it as a quotient. 00:01:57.500 --> 00:02:00.330 There are a few other indeterminate forms that I'd 00:02:00.330 --> 00:02:01.990 like to mention. 00:02:01.990 --> 00:02:04.460 So in particular, there are three other indeterminate 00:02:04.460 --> 00:02:07.890 forms not, different from these. 00:02:07.890 --> 00:02:10.900 And then I'll give you an example of one of them to 00:02:10.900 --> 00:02:12.300 solve a problem. 00:02:12.300 --> 00:02:15.690 so the three other indeterminate forms, there are 00:02:15.690 --> 00:02:18.190 two more that are an exponential indeterminate 00:02:18.190 --> 00:02:21.100 form, and there's one sort of outlier. 00:02:21.100 --> 00:02:26.410 So one of the other indeterminate forms is 00:02:26.410 --> 00:02:28.750 infinity to the 0. 00:02:28.750 --> 00:02:32.440 So when you have an exponential expression where 00:02:32.440 --> 00:02:36.170 the base is getting very very large, and the exponent is 00:02:36.170 --> 00:02:38.683 going to 0, well the base getting large makes the whole 00:02:38.683 --> 00:02:42.040 fraction want to be-- sorry-- the whole expression want to 00:02:42.040 --> 00:02:44.930 be large, whereas the exponent going to 0 makes the whole 00:02:44.930 --> 00:02:46.730 expression want to get closer to 1. 00:02:46.730 --> 00:02:49.160 And so those two forces are in tension, and you can end up 00:02:49.160 --> 00:02:53.510 with limits that equal any value when you have a limit 00:02:53.510 --> 00:02:55.580 like this, infinity to the 0. 00:02:55.580 --> 00:03:00.410 Another similar one is 1 to the infinity. 00:03:00.410 --> 00:03:01.770 And this is the one I'll give you an 00:03:01.770 --> 00:03:03.250 exercise on in a minute. 00:03:03.250 --> 00:03:06.450 So when you have a limit of the form 1 to the infinity, so 00:03:06.450 --> 00:03:09.510 something to the something where the base is going to 1 00:03:09.510 --> 00:03:12.760 and the exponent is going to infinity, the base going to 1 00:03:12.760 --> 00:03:14.480 makes this whole thing want to go to 1. 00:03:14.480 --> 00:03:17.840 The exponent to infinity makes this whole thing want to 00:03:17.840 --> 00:03:21.480 either blow up if it's a little bigger than 1, or get 00:03:21.480 --> 00:03:23.580 really small if it's a little smaller than 1. 00:03:23.580 --> 00:03:26.420 So you have, again, a tension there, and the result is 00:03:26.420 --> 00:03:27.730 indeterminate. 00:03:27.730 --> 00:03:31.400 The one sort of unusual one that you have is also-- 00:03:31.400 --> 00:03:34.610 which I'm not going to talk much more about-- is infinity 00:03:34.610 --> 00:03:35.860 minus infinity. 00:03:38.840 --> 00:03:40.630 So when you have two very large things, so here I mean 00:03:40.630 --> 00:03:43.830 either positive infinity minus positive infinity, or negative 00:03:43.830 --> 00:03:45.500 infinity minus negative infinity. 00:03:45.500 --> 00:03:48.430 When you have two things that are both getting very large, 00:03:48.430 --> 00:03:50.640 their difference could also be getting very large, or it 00:03:50.640 --> 00:03:52.560 could be getting very small, or could be 00:03:52.560 --> 00:03:55.380 doing anything in between. 00:03:55.380 --> 00:03:57.820 So that's also an indeterminate form. 00:03:57.820 --> 00:04:01.400 So these are the seven indeterminate forms. When you 00:04:01.400 --> 00:04:04.920 have a quotient, you can always apply l'Hopital's rule. 00:04:04.920 --> 00:04:07.580 When you have a product, you can always rewrite it as a 00:04:07.580 --> 00:04:14.930 quotient, by writing for example, x squared plus 1 00:04:14.930 --> 00:04:16.180 times-- well, let's see-- 00:04:18.230 --> 00:04:22.010 e to the minus x. 00:04:22.010 --> 00:04:23.110 That's a product. 00:04:23.110 --> 00:04:26.710 And you could always rewrite it as a quotient, for example, 00:04:26.710 --> 00:04:32.710 e to the minus x over 1 over x squared plus 1. 00:04:32.710 --> 00:04:35.350 There might even be a smarter way to rewrite this product as 00:04:35.350 --> 00:04:36.600 a quotient. 00:04:38.620 --> 00:04:38.674 But, OK. 00:04:38.674 --> 00:04:41.340 And so when you have an exponential, as in this case, 00:04:41.340 --> 00:04:44.790 you can use this rewriting in base e trick to turn it into a 00:04:44.790 --> 00:04:48.270 product, which you can then turn into a quotient. 00:04:48.270 --> 00:04:50.780 For the difference case, the reason I say it's unusual, is 00:04:50.780 --> 00:04:55.140 just that there's not a good, general method for working 00:04:55.140 --> 00:04:55.710 with these. 00:04:55.710 --> 00:04:59.590 There are a lot of special cases. 00:04:59.590 --> 00:04:59.850 And you sort of have to-- or what I mean is you have to 00:04:59.850 --> 00:05:01.890 analyze them on a case by case basis. 00:05:01.890 --> 00:05:06.120 There are different sort of techniques that will work. 00:05:06.120 --> 00:05:10.110 So let me give you an example of one of these 1 to the 00:05:10.110 --> 00:05:11.960 infinity kinds. 00:05:11.960 --> 00:05:13.240 So why don't we come over here. 00:05:13.240 --> 00:05:18.840 So compute the limit as x goes to 0 from the right of 1 plus 00:05:18.840 --> 00:05:21.170 3x to the 10 divided by x. 00:05:21.170 --> 00:05:25.180 So we see as x is going to 0, this base is going to 1. 00:05:25.180 --> 00:05:27.840 And so that makes this whole expression want to be close to 00:05:27.840 --> 00:05:33.160 1, whereas this exponent is going to infinity, since this 00:05:33.160 --> 00:05:35.570 is just a little bit bigger than 1, that makes the whole 00:05:35.570 --> 00:05:40.090 expression want to be big when the exponent is big. 00:05:40.090 --> 00:05:43.950 So you have a tension here between the base going to 1 00:05:43.950 --> 00:05:45.450 and the exponent going to infinity. 00:05:45.450 --> 00:05:48.460 So the question is, try and actually compute this limit. 00:05:48.460 --> 00:05:50.270 So why don't you pause the video, take a couple of 00:05:50.270 --> 00:05:52.360 minutes to work on this question, come back, and we 00:05:52.360 --> 00:05:53.610 can work on it together. 00:06:03.440 --> 00:06:04.050 Welcome back. 00:06:04.050 --> 00:06:07.250 Hopefully you had some luck working on this problem. 00:06:07.250 --> 00:06:11.270 As we said, this is a limit of an indeterminate form of the 1 00:06:11.270 --> 00:06:12.420 to infinity type. 00:06:12.420 --> 00:06:14.510 As one of the three exponential types of 00:06:14.510 --> 00:06:18.240 indeterminate forms, a really promising first step almost 00:06:18.240 --> 00:06:22.090 every time is to rewrite this as a exponential expression 00:06:22.090 --> 00:06:23.650 with the base e. 00:06:23.650 --> 00:06:25.590 So we just do, so first we just do an algebraic 00:06:25.590 --> 00:06:28.520 manipulation on the thing we're taking the limit of, and 00:06:28.520 --> 00:06:31.460 then often, very often, that simplifies it into something 00:06:31.460 --> 00:06:34.040 that we can actually compute the limit of. 00:06:34.040 --> 00:06:39.150 So in particular, we have that 1 plus 3x, if we want to write 00:06:39.150 --> 00:06:41.600 this in exponential form. 00:06:41.600 --> 00:06:46.850 This is equal to e to the ln of 1 plus 3x. 00:06:46.850 --> 00:06:50.560 This is true of any positive number. 00:06:50.560 --> 00:06:54.470 Any positive number is e to the ln of it, because e and 00:06:54.470 --> 00:06:57.480 log are inverse functions. 00:06:57.480 --> 00:07:00.120 So OK, so we right it like this. 00:07:00.120 --> 00:07:07.210 And so that means that 1 plus 3x to the 10 over x is equal 00:07:07.210 --> 00:07:16.550 to e to the ln of 1 plus 3x to the 10 over x. 00:07:16.550 --> 00:07:19.970 And now you can use your exponent rules. 00:07:19.970 --> 00:07:27.960 So this is equal to e to the ln of 1 plus 3x 00:07:27.960 --> 00:07:32.510 times 10 over x. 00:07:32.510 --> 00:07:37.430 So our original expression is equal to this down here. 00:07:37.430 --> 00:07:39.750 So the limit of our original expression is equal to the 00:07:39.750 --> 00:07:42.690 limit of this one. 00:07:42.690 --> 00:07:45.350 The other thing to notice is that because exponentiation is 00:07:45.350 --> 00:07:48.160 a nice, continuous function, in order to compute this 00:07:48.160 --> 00:07:52.410 limit, or the limit of this expression, it suffices-- 00:07:52.410 --> 00:07:57.480 just with base a constant, e-- it suffices to compute the 00:07:57.480 --> 00:07:59.740 limit of the exponent. 00:07:59.740 --> 00:08:02.020 All right, so let's do that. 00:08:02.020 --> 00:08:04.070 Let's compute the limit of the exponent. 00:08:04.070 --> 00:08:08.950 So we have the limit, so it has to be x going to the same 00:08:08.950 --> 00:08:16.400 place, which in this case is to 0 from the right of ln of 1 00:08:16.400 --> 00:08:21.500 plus 3x times 10 over x. 00:08:21.500 --> 00:08:22.205 Well, this is a product. 00:08:22.205 --> 00:08:23.740 It came to us as a product. 00:08:23.740 --> 00:08:27.540 But there's an obvious way to rewrite this as a quotient. 00:08:27.540 --> 00:08:30.150 I should say it's an indeterminate product. 00:08:30.150 --> 00:08:34.250 As x goes to 0, this goes to ln of 1, which is 0. 00:08:34.250 --> 00:08:37.390 Whereas this goes to infinity, positive infinity since we're 00:08:37.390 --> 00:08:38.500 coming from the right. 00:08:38.500 --> 00:08:40.790 So this is a 0 times positive infinity form. 00:08:40.790 --> 00:08:42.960 So it is an indeterminate limit-- or sorry-- an 00:08:42.960 --> 00:08:45.840 indeterminate product. 00:08:48.870 --> 00:08:52.040 And, right, and there's an obvious way to rewrite this as 00:08:52.040 --> 00:08:55.950 an indeterminate quotient, which is to rewrite it as the 00:08:55.950 --> 00:09:08.100 limit as x goes 0 plus of ten ln of 1 plus 3x divided by x 00:09:08.100 --> 00:09:09.440 as x goes 0. 00:09:09.440 --> 00:09:15.680 So now, this is a limit where a it's an infinity-- sorry-- 00:09:15.680 --> 00:09:18.610 a 0 over 0. 00:09:18.610 --> 00:09:19.370 So OK, so good. 00:09:19.370 --> 00:09:20.900 So now we can apply l'Hopital's rule. 00:09:20.900 --> 00:09:26.250 So by l'Hopital's rule, this is equal to the limit as x 00:09:26.250 --> 00:09:29.420 goes to 0 on the right. 00:09:29.420 --> 00:09:33.790 Of, well we apply l'Hopital's rule on the top, we get 10 00:09:33.790 --> 00:09:40.460 over 1 plus 3x, by the chain rule, times 3. 00:09:40.460 --> 00:09:41.670 And on the bottom, we just get one. 00:09:41.670 --> 00:09:48.840 So now, OK, so this is true provided this 00:09:48.840 --> 00:09:51.440 second limit exists. 00:09:51.440 --> 00:09:54.500 And the second limit is no longer indeterminate. 00:09:54.500 --> 00:09:55.780 It's easy to see what it is. 00:09:55.780 --> 00:09:57.990 You just plug in x equals 0, and we get that 00:09:57.990 --> 00:10:04.390 this is equal to 30. 00:10:04.390 --> 00:10:10.850 OK so this limit is equal to 30, but this isn't the limit 00:10:10.850 --> 00:10:12.880 that we started out wanting to compute. 00:10:12.880 --> 00:10:17.330 The limit we started out wanting to compute is e to the 00:10:17.330 --> 00:10:20.990 ln of 1 plus 3x times 10 over x. 00:10:20.990 --> 00:10:23.340 So it's e to the this. 00:10:23.340 --> 00:10:33.690 So our original limit as x goes to 0 from the right of 1 00:10:33.690 --> 00:10:41.200 plus 3x to the 10 over x is equal to e to the thirtieth 00:10:41.200 --> 00:10:44.989 power, which is pretty huge. 00:10:44.989 --> 00:10:53.750 OK, So that's how we-- right-- and OK, so that's the answer 00:10:53.750 --> 00:10:54.950 to our question. 00:10:54.950 --> 00:10:58.750 So we took our original limit, it was this indeterminate 00:10:58.750 --> 00:11:00.020 exponential form. 00:11:00.020 --> 00:11:02.790 So what we do to it is, when you have an indeterminate 00:11:02.790 --> 00:11:06.490 exponential form, you do this rewriting as an exponential in 00:11:06.490 --> 00:11:08.610 the base of e trick, and then you pass the 00:11:08.610 --> 00:11:10.040 limit into the exponent. 00:11:10.040 --> 00:11:12.460 Because e is now a nice constant, life is simple-- 00:11:14.980 --> 00:11:16.720 life is simpler I should say-- 00:11:16.720 --> 00:11:18.960 you pass the limit into the constant, then you have an 00:11:18.960 --> 00:11:22.060 indeterminate ratio, indeterminate product, which 00:11:22.060 --> 00:11:24.300 you rewrite as an indeterminate ratio on which 00:11:24.300 --> 00:11:26.020 you can then apply l'Hopital's rule. 00:11:26.020 --> 00:11:29.730 Or possibly, you know, you rewrite it as a product. 00:11:29.730 --> 00:11:34.270 Then it's easy to see what the value is. 00:11:34.270 --> 00:11:37.700 So, all right, so that's how we deal with limits of 00:11:37.700 --> 00:11:41.050 indeterminate exponential forms. And I'll end there.