WEBVTT 00:00:07.580 --> 00:00:08.060 PROFESSOR: OK. 00:00:08.060 --> 00:00:12.480 I promised a video about limits and continuous 00:00:12.480 --> 00:00:16.370 functions and here it is. 00:00:16.370 --> 00:00:23.530 So I'll begin with the most basic idea and with a picture 00:00:23.530 --> 00:00:28.730 instead of definition in symbols first. So the most 00:00:28.730 --> 00:00:31.710 basic idea is that I have a bunch of numbers-- 00:00:31.710 --> 00:00:34.330 let's make them positive numbers-- 00:00:34.330 --> 00:00:37.440 and I want to know, what does it mean for them 00:00:37.440 --> 00:00:40.220 to approach a limit-- 00:00:40.220 --> 00:00:41.580 capital A-- 00:00:41.580 --> 00:00:46.810 as I go out this sequence of numbers, a1, a2, a3, a4. 00:00:46.810 --> 00:00:50.100 And let me say right away, the first four numbers, the first 00:00:50.100 --> 00:00:54.320 million numbers, make no difference about the limit. 00:00:54.320 --> 00:00:57.080 So here's what it means. 00:00:57.080 --> 00:01:00.970 For example, there's a equals 7, let's say. 00:01:00.970 --> 00:01:05.190 What does it mean for these numbers to approach 7? 00:01:05.190 --> 00:01:13.840 It means that if I take any thin little space around a, 00:01:13.840 --> 00:01:18.450 above and below, the numbers can start out whatever. 00:01:18.450 --> 00:01:21.170 They can go in there, they could go out, they could come 00:01:21.170 --> 00:01:25.880 back, whatever, they could grow way big, way small. 00:01:25.880 --> 00:01:32.700 But in the end, beyond some point, eventually, they have 00:01:32.700 --> 00:01:38.310 to get in that slit and stay in there. 00:01:38.310 --> 00:01:41.680 And the slit, then, could be smaller. 00:01:41.680 --> 00:01:45.220 And then they would have to get into that smaller slit and 00:01:45.220 --> 00:01:46.470 stay there. 00:01:46.470 --> 00:01:50.510 So that's what it means for the numbers to approach A, 00:01:50.510 --> 00:01:56.660 that eventually after any number of jogs around, they 00:01:56.660 --> 00:01:59.660 get inside and they stay there, however 00:01:59.660 --> 00:02:01.370 thin that slit is. 00:02:01.370 --> 00:02:04.180 A slight difference when a is 0, because 00:02:04.180 --> 00:02:06.870 the numbers are positive. 00:02:06.870 --> 00:02:10.110 They're coming down, they get in. 00:02:10.110 --> 00:02:12.920 And again, they must stay in. 00:02:12.920 --> 00:02:18.650 And then, again, I'm going to make the band tighter, and 00:02:18.650 --> 00:02:20.950 they have to get into that and stay there. 00:02:20.950 --> 00:02:25.020 And what does it mean for the numbers to approach infinity? 00:02:25.020 --> 00:02:27.270 That means that whatever-- 00:02:27.270 --> 00:02:31.080 so this is often called epsilon. 00:02:31.080 --> 00:02:34.820 I'll use that Greek letter epsilon as 00:02:34.820 --> 00:02:36.290 a very small number. 00:02:36.290 --> 00:02:40.100 So this would be A minus epsilon, and this would be A 00:02:40.100 --> 00:02:41.500 plus epsilon. 00:02:41.500 --> 00:02:44.010 And then the epsilon could be made smaller. 00:02:44.010 --> 00:02:45.770 And now here is some big number, 00:02:45.770 --> 00:02:48.800 like even 1 over epsilon. 00:02:48.800 --> 00:02:50.890 So that's a giant number. 00:02:50.890 --> 00:02:55.810 And the limit is infinity if, again, they can dodge around 00:02:55.810 --> 00:02:58.060 for a while, they can go down, they can go up. 00:02:58.060 --> 00:03:03.160 But eventually, they must get above that 00:03:03.160 --> 00:03:05.230 line and stay there. 00:03:05.230 --> 00:03:08.190 And if I move the line up further, they have to get 00:03:08.190 --> 00:03:11.770 above that line for me to say that the limit is-- 00:03:11.770 --> 00:03:15.460 so I have these possible limits. 00:03:15.460 --> 00:03:20.180 Infinite, some positive, ordinary number, and 0. 00:03:20.180 --> 00:03:22.630 Those are possible limits. 00:03:22.630 --> 00:03:27.570 But of course many sequences have no limit at all, like 00:03:27.570 --> 00:03:31.280 sine of n, it will just bounce around, cosine n-- 00:03:31.280 --> 00:03:32.700 many, many things. 00:03:32.700 --> 00:03:33.590 OK. 00:03:33.590 --> 00:03:42.280 So I think that the way to get the idea, use the idea, is to 00:03:42.280 --> 00:03:46.880 ask some questions about limits. 00:03:46.880 --> 00:03:52.570 And we'll see that usually the answer is yes, OK, no problem. 00:03:52.570 --> 00:03:57.900 But once in a while, for certain limits are dangerous. 00:03:57.900 --> 00:04:03.250 So really always mathematicians are looking for 00:04:03.250 --> 00:04:07.370 what's special, what unusual thing could happen? 00:04:07.370 --> 00:04:12.980 Because the truth is, limits are ordinarily rather boring. 00:04:12.980 --> 00:04:19.220 If the an's approach 7 and the bn's approach 4, so the a's 00:04:19.220 --> 00:04:22.220 get close to 7 and the b's get close to 4, then their 00:04:22.220 --> 00:04:26.020 differences will get close to 7 minus 4. 00:04:28.780 --> 00:04:32.930 But is there any case in which that could fail? 00:04:32.930 --> 00:04:38.940 Is there any case among these in which we could not know 00:04:38.940 --> 00:04:42.140 what the limit was, and it might not exist, or it might 00:04:42.140 --> 00:04:47.040 be like any number? 00:04:47.040 --> 00:04:50.400 And I think that can happen in this, so I've got four 00:04:50.400 --> 00:04:55.570 different questions here, getting more 00:04:55.570 --> 00:04:57.430 interesting as we go down. 00:04:57.430 --> 00:05:01.350 In the first one, I can see only one problem. 00:05:01.350 --> 00:05:06.770 If the a's approach infinity, so they get very big, and the 00:05:06.770 --> 00:05:11.860 b's also approach infinity, get very big, so capital A and 00:05:11.860 --> 00:05:21.010 capital B become formally infinity minus infinity, and 00:05:21.010 --> 00:05:23.180 we don't know the answer there. 00:05:23.180 --> 00:05:24.890 That has no meaning. 00:05:24.890 --> 00:05:28.780 So this'll be my little list of danger. 00:05:32.680 --> 00:05:35.660 I mean, it's not like skydiving, but for a 00:05:35.660 --> 00:05:38.260 mathematician this is high risk. 00:05:38.260 --> 00:05:40.760 OK, so how could this happen? 00:05:40.760 --> 00:05:44.890 Well, the an might be n squared. 00:05:44.890 --> 00:05:47.230 And the bn might be only n. 00:05:47.230 --> 00:05:49.230 Right? 00:05:49.230 --> 00:05:52.320 So they're both going to infinity, n squared and n. 00:05:52.320 --> 00:05:53.820 But n squared is going faster. 00:05:53.820 --> 00:05:55.590 It's like a race. 00:05:55.590 --> 00:05:59.590 n squared will win, and the difference between them will 00:05:59.590 --> 00:06:02.670 actually grow faster and faster. 00:06:02.670 --> 00:06:05.410 Or they could go to infinity together. 00:06:05.410 --> 00:06:10.110 an and bn could both be n, both headed for infinity. 00:06:10.110 --> 00:06:14.640 The differences would be 0 all along, n minus n. 00:06:14.640 --> 00:06:17.550 So the limit of the difference would be 0 minus 0. 00:06:17.550 --> 00:06:21.060 So this could be 0, but it could be infinity, it could be 00:06:21.060 --> 00:06:23.410 minus infinity, it could be anything. 00:06:23.410 --> 00:06:25.686 Any limit is possible there. 00:06:25.686 --> 00:06:29.610 Do you see that there is a case-- 00:06:29.610 --> 00:06:32.800 it's sort of a special case, because it only happens when 00:06:32.800 --> 00:06:34.840 these limits are infinite-- 00:06:34.840 --> 00:06:41.110 but now it's sort of OK to look at each-- 00:06:41.110 --> 00:06:43.230 let me look at number two. 00:06:43.230 --> 00:06:46.240 How about multiplication? 00:06:46.240 --> 00:06:49.850 If I multiply a bunch of numbers that are headed for 7 00:06:49.850 --> 00:06:52.540 and a bunch of numbers that are headed for 4, their 00:06:52.540 --> 00:06:55.540 product is going to head for 28. 00:06:55.540 --> 00:06:58.650 This will be true. 00:06:58.650 --> 00:07:01.960 When could it fail? 00:07:01.960 --> 00:07:06.410 Well, again, it's going to be extreme cases, because if I 00:07:06.410 --> 00:07:09.270 have ordinary numbers for A and B like 7 and 00:07:09.270 --> 00:07:11.690 4, there's no doubt. 00:07:11.690 --> 00:07:19.440 But look at the extreme case of when A 00:07:19.440 --> 00:07:22.685 is 0 and B is infinite. 00:07:26.930 --> 00:07:29.690 So the an's are headed for 0. 00:07:29.690 --> 00:07:32.600 The bn's are getting bigger and bigger, the an's are 00:07:32.600 --> 00:07:37.000 getting small as we go far enough out. 00:07:37.000 --> 00:07:37.690 OK. 00:07:37.690 --> 00:07:42.490 In that case, well, again it's a race. 00:07:42.490 --> 00:07:46.680 The an's might be 1 over n squared, and the 00:07:46.680 --> 00:07:49.250 bn's might be n. 00:07:49.250 --> 00:07:53.840 So this would be n over n squared, and 00:07:53.840 --> 00:07:55.790 that would go to 0. 00:07:55.790 --> 00:08:03.960 But if I reverse those I could have n squared times 1 over n. 00:08:03.960 --> 00:08:06.780 The product could get bigger, or the product could-- 00:08:06.780 --> 00:08:08.480 all possibilities. 00:08:08.480 --> 00:08:09.940 All possibilities there. 00:08:09.940 --> 00:08:12.250 So I cannot know what that one is. 00:08:12.250 --> 00:08:15.080 0 times infinity is meaningless. 00:08:15.080 --> 00:08:15.810 OK. 00:08:15.810 --> 00:08:17.680 What about number three? 00:08:20.360 --> 00:08:24.610 The danger increases as soon as we start dividing. 00:08:24.610 --> 00:08:27.080 I made the b's positive, but I don't know if 00:08:27.080 --> 00:08:28.650 capital B is positive. 00:08:28.650 --> 00:08:29.960 So the danger-- 00:08:29.960 --> 00:08:35.110 and, in fact, the most important case for calculus-- 00:08:35.110 --> 00:08:38.480 is 0 over 0. 00:08:38.480 --> 00:08:43.210 If the a's go to 0 and the b's go to 0, I can't tell what 00:08:43.210 --> 00:08:49.430 their ratio goes to, because it depends how fast they go. 00:08:49.430 --> 00:08:53.950 If the a's go quickly to 0 and the b's are rather slow 00:08:53.950 --> 00:08:55.260 getting there-- 00:08:55.260 --> 00:09:00.060 in other words the b's would be a lot bigger than the a's 00:09:00.060 --> 00:09:02.050 even though both are going to 0-- 00:09:02.050 --> 00:09:04.730 then that fraction would be small. 00:09:04.730 --> 00:09:07.750 But if I reverse them, the fraction would be large. 00:09:07.750 --> 00:09:11.180 So I think 0 over 0 is a danger. 00:09:11.180 --> 00:09:15.290 I think there's another danger here. 00:09:15.290 --> 00:09:17.540 Yeah, maybe infinity over infinity. 00:09:17.540 --> 00:09:25.810 Again, that's a race that we can't tell, until we know 00:09:25.810 --> 00:09:29.220 details about the sequences, who's going to win. 00:09:29.220 --> 00:09:32.530 If they an's go off to infinity and the bn's go off 00:09:32.530 --> 00:09:35.910 to infinity, ah, a very important case. 00:09:35.910 --> 00:09:39.710 The ratio could be 1 all along. 00:09:39.710 --> 00:09:43.420 The a's and b's could be the same, headed for infinity. 00:09:43.420 --> 00:09:48.380 Or the an's might be squaring the b's and going up faster, 00:09:48.380 --> 00:09:51.195 or the square root of the b's and going slower. 00:09:54.710 --> 00:09:58.510 So again, infinity over infinity, we can't-- 00:09:58.510 --> 00:10:04.250 0 over infinity, if the a's are headed for 0 and the b's 00:10:04.250 --> 00:10:08.400 are headed big, then that ratio is going to be small and 00:10:08.400 --> 00:10:09.310 head for 0. 00:10:09.310 --> 00:10:11.430 0 over infinity, I'm OK with. 00:10:11.430 --> 00:10:13.640 Call it 0. 00:10:13.640 --> 00:10:15.350 Well, I don't know if that's legal, but 00:10:15.350 --> 00:10:16.870 anyway, let me do it. 00:10:16.870 --> 00:10:20.060 All right, last one of this kind, just for 00:10:20.060 --> 00:10:21.520 practice about limits. 00:10:21.520 --> 00:10:24.990 Again, you see what I'm constantly doing is thinking 00:10:24.990 --> 00:10:32.010 of examples that simply show that I can't 00:10:32.010 --> 00:10:33.370 be sure of the limit. 00:10:33.370 --> 00:10:38.600 So here normally I could be sure, if this is headed for 7 00:10:38.600 --> 00:10:42.340 to the fourth, that'll be the limit, whatever that is, 49 00:10:42.340 --> 00:10:47.510 squared, 2401, or something. 00:10:47.510 --> 00:10:49.680 But if-- 00:10:49.680 --> 00:10:52.160 now when could it go wrong? 00:10:52.160 --> 00:10:53.990 Here's an interesting case. 00:10:53.990 --> 00:10:58.420 So this is my list of danger, and I think I'm in danger if 00:10:58.420 --> 00:11:01.000 they both go to 0. 00:11:01.000 --> 00:11:05.400 0 to the 0-th power, I don't know what that is. 00:11:05.400 --> 00:11:09.210 And actually, I don't know all the possibilities here. 00:11:09.210 --> 00:11:19.020 I can see one way would be let's suppose the b's were 00:11:19.020 --> 00:11:22.340 actually 0, or practically. 00:11:22.340 --> 00:11:26.550 Then things to the 0 power are 1. 00:11:26.550 --> 00:11:32.180 So I could get the answer 1 here by fixing the b's at 0 00:11:32.180 --> 00:11:35.860 and letting these guys, they would all be to the 0 power, 00:11:35.860 --> 00:11:37.440 so they would all be 1. 00:11:37.440 --> 00:11:38.945 And in the limit, I would have 1. 00:11:42.270 --> 00:11:44.440 But I could also do it differently. 00:11:44.440 --> 00:11:49.610 I could fix these at 0 and let these guys get smaller. 00:11:49.610 --> 00:11:52.220 Then I would have 0 to powers. 00:11:52.220 --> 00:11:54.610 And zero to any power is 0. 00:11:54.610 --> 00:11:56.400 You see my little problem here? 00:11:56.400 --> 00:11:58.190 Let me write my little problem here. 00:11:58.190 --> 00:12:03.620 My problem is that a to the 0 power would be 1, but 0 to the 00:12:03.620 --> 00:12:09.435 a-th power would be 0, or 0 to the b-th, maybe I should say. 00:12:12.870 --> 00:12:16.840 So if I'm in this situation and the a is shrinking to 0, I 00:12:16.840 --> 00:12:18.680 still have a limit of 1's. 00:12:18.680 --> 00:12:21.660 But if I'm in this situation and the b's are headed for 0, 00:12:21.660 --> 00:12:22.740 I have a limit of 0's. 00:12:22.740 --> 00:12:26.540 And maybe you could get 1/2, I don't know how. 00:12:26.540 --> 00:12:31.370 And you have to allow me-- because I have to finish this 00:12:31.370 --> 00:12:34.540 list, and I only have one more to tell you-- 00:12:34.540 --> 00:12:39.120 that another case, a very interesting type of calculus 00:12:39.120 --> 00:12:44.020 case is the case where the a's go to 1 and 00:12:44.020 --> 00:12:45.345 the b's go to infinity. 00:12:48.460 --> 00:12:53.640 I don't know if you remember that this actually happened in 00:12:53.640 --> 00:13:00.140 the lecture on e, the number that comes in e to the x, the 00:13:00.140 --> 00:13:01.670 great number of calculus. 00:13:01.670 --> 00:13:03.440 Do you remember that? 00:13:03.440 --> 00:13:06.920 So I'm going to talk a little bit about the a's going to 1 00:13:06.920 --> 00:13:10.120 and the b's blowing up. 00:13:10.120 --> 00:13:12.960 So I'm getting things that are very near 1, but I'm taking 00:13:12.960 --> 00:13:14.830 many, many more of them. 00:13:14.830 --> 00:13:17.350 And I believe that I can get all kinds of 00:13:17.350 --> 00:13:19.710 different limits there. 00:13:19.710 --> 00:13:22.180 I believe I can get all kinds of different limits. 00:13:22.180 --> 00:13:23.120 Do you just-- 00:13:23.120 --> 00:13:26.180 maybe on this next board. 00:13:26.180 --> 00:13:32.710 And then I promise to come back to the heart of the 00:13:32.710 --> 00:13:36.270 subject of limits and continuous functions. 00:13:36.270 --> 00:13:44.920 But I just think that one, the famous case of this one, was 1 00:13:44.920 --> 00:13:47.060 plus 1 over n. 00:13:47.060 --> 00:13:51.200 That's the a's, and that approaches what limit? 00:13:51.200 --> 00:13:52.810 One. 00:13:52.810 --> 00:13:56.120 The b's I'm going to take as n. 00:13:56.120 --> 00:13:58.460 So the b's are going to infinity. 00:13:58.460 --> 00:14:01.240 So I'm discussing this case here. 00:14:01.240 --> 00:14:03.150 So that's a case where this goes to 1, 00:14:03.150 --> 00:14:04.260 this goes to infinity. 00:14:04.260 --> 00:14:07.992 I had an email this week saying, wait a minute, I've 00:14:07.992 --> 00:14:10.540 got a little problem here, because I know 1 to the 00:14:10.540 --> 00:14:14.060 infinity is e. 00:14:14.060 --> 00:14:15.840 1 to the infinity is 3. 00:14:15.840 --> 00:14:23.050 Well, that's because it's true that that number approaches e. 00:14:23.050 --> 00:14:27.640 That's one of the many remarkable ways to produce the 00:14:27.640 --> 00:14:30.640 number e, the 2.7-something. 00:14:30.640 --> 00:14:37.210 But that's because the race between this and this was so 00:14:37.210 --> 00:14:38.460 evenly balanced. 00:14:41.140 --> 00:14:46.360 If I took these closer and closer to 1, like n squared, 00:14:46.360 --> 00:14:48.420 what would happen then? 00:14:48.420 --> 00:14:52.070 Then I'm taking numbers very, very near 1, I'm taking a 00:14:52.070 --> 00:14:55.530 power, but these are sort of near, those would approach-- 00:14:55.530 --> 00:14:58.660 would you like to guess? 00:14:58.660 --> 00:15:00.820 One. 00:15:00.820 --> 00:15:03.990 These are so close to 1 that taking the nth power doesn't 00:15:03.990 --> 00:15:05.690 move them far. 00:15:05.690 --> 00:15:12.070 And you can guess that I could get infinity too, by taking n 00:15:12.070 --> 00:15:14.860 not still close to 1 and taking some big 00:15:14.860 --> 00:15:18.950 power like n squared. 00:15:18.950 --> 00:15:21.950 Now I have things close to 1, but I'm taking so many of them 00:15:21.950 --> 00:15:23.200 that it would blow up. 00:15:25.820 --> 00:15:29.120 So again i think--So those are all cases where in the limit, 00:15:29.120 --> 00:15:31.780 I have 1, in the limit, I have infinity. 00:15:31.780 --> 00:15:41.620 But that combination 1 to the increasingly high powers can 00:15:41.620 --> 00:15:43.790 do different things. 00:15:43.790 --> 00:15:49.790 This was my little idea to show you the risky cases. 00:15:49.790 --> 00:15:50.820 OK. 00:15:50.820 --> 00:15:57.960 But actually, 0 over 0, that's what calculus is 00:15:57.960 --> 00:16:00.290 always doing, right? 00:16:00.290 --> 00:16:04.730 Because that's exactly what we have when we have a delta f 00:16:04.730 --> 00:16:08.310 over a delta x, a delta y over a delta x. 00:16:08.310 --> 00:16:14.010 They're both approaching 0 and we get a definite slope when 00:16:14.010 --> 00:16:17.420 the ratio goes to a good number. 00:16:17.420 --> 00:16:18.340 OK. 00:16:18.340 --> 00:16:21.980 So can I discuss 0 over 0? 00:16:21.980 --> 00:16:25.530 All right, phooey on this one. 00:16:25.530 --> 00:16:26.250 OK. 00:16:26.250 --> 00:16:32.560 So I now want to speak about the case when f 00:16:32.560 --> 00:16:33.810 of x goes to 0. 00:16:37.140 --> 00:16:41.080 Let's say f of x goes to 0 as x goes to 0. 00:16:41.080 --> 00:16:45.570 So there'll be an if here. 00:16:45.570 --> 00:16:48.680 I have to say what that means. 00:16:48.680 --> 00:16:55.430 And then I'm also going to have some g of x going to 0 as 00:16:55.430 --> 00:16:57.760 x goes to 0. 00:16:57.760 --> 00:17:02.210 OK, so both functions are decreasing. 00:17:02.210 --> 00:17:07.530 And my question, let me ask the question first, what about 00:17:07.530 --> 00:17:09.740 f of x over g of x? 00:17:12.339 --> 00:17:14.150 What does that do? 00:17:14.150 --> 00:17:22.499 And of course, just as I said up there, I can't tell yet. 00:17:25.420 --> 00:17:27.510 I have to know the f and the g. 00:17:27.510 --> 00:17:31.190 It's a race to 0, and I have to know who's the winner and 00:17:31.190 --> 00:17:32.710 by how much. 00:17:32.710 --> 00:17:36.450 But first, I'd better say what does it mean for a function to 00:17:36.450 --> 00:17:38.900 go to 0 as x goes to 0. 00:17:38.900 --> 00:17:40.230 Well, you know. 00:17:40.230 --> 00:17:44.140 Let me draw a graph of this function. 00:17:44.140 --> 00:17:45.690 OK, I'll just draw it. 00:17:45.690 --> 00:17:47.390 So f of x is going to 0. 00:17:47.390 --> 00:17:54.140 So here is x, and I'm going to graph f of x, and here is 0. 00:17:54.140 --> 00:18:00.850 So as x is coming down to 0, my f of x is also coming to 0. 00:18:00.850 --> 00:18:05.350 So it could come like so. 00:18:05.350 --> 00:18:10.460 That's a pretty sensible, smooth, nice approach to 0. 00:18:10.460 --> 00:18:14.300 That could be my f of x. 00:18:14.300 --> 00:18:21.400 And it may be a g of x is smaller, but also approaching 00:18:21.400 --> 00:18:23.590 0 in a nice, smooth way. 00:18:23.590 --> 00:18:29.720 This is a case where you can see those, 00:18:29.720 --> 00:18:32.020 as x goes that way-- 00:18:32.020 --> 00:18:34.680 maybe the arrow should be going that way, because x is 00:18:34.680 --> 00:18:36.590 going to 0-- 00:18:36.590 --> 00:18:41.690 my f of x is getting smaller, my g of x is getting smaller. 00:18:41.690 --> 00:18:46.090 And I'll say exactly what that means, but you 00:18:46.090 --> 00:18:47.320 know what it means. 00:18:47.320 --> 00:18:51.750 It means that if I put a little, like these lines, if I 00:18:51.750 --> 00:18:56.160 put a little band there, it gets into that band. 00:18:56.160 --> 00:18:59.610 Actually, g will get into the band sooner. 00:18:59.610 --> 00:19:03.300 But then f will safely get into the band. 00:19:03.300 --> 00:19:08.760 Now, the question is what about f of x over g of x? 00:19:08.760 --> 00:19:10.510 OK. 00:19:10.510 --> 00:19:11.660 Can we say? 00:19:11.660 --> 00:19:16.840 Now, I'm going to suppose that f of x has a 00:19:16.840 --> 00:19:20.100 definite slope, s. 00:19:20.100 --> 00:19:22.975 And this one has a definite slope, t. 00:19:26.750 --> 00:19:26.998 In other words, I am going to suppose-- 00:19:26.998 --> 00:19:32.950 Here look, this is called, named after a French guy, 00:19:32.950 --> 00:19:38.470 L'Hopital, the hospital rule. 00:19:38.470 --> 00:19:49.320 OK, so it's just a little trick, because this comes up 00:19:49.320 --> 00:19:54.320 of what's happening in this race to 0. 00:19:54.320 --> 00:20:01.280 And the natural idea is that f of x is really, since f is 0 00:20:01.280 --> 00:20:05.820 there, and I'm really just going a little way. 00:20:05.820 --> 00:20:11.170 So maybe I call that delta x, just to emphasize that I'm 00:20:11.170 --> 00:20:14.220 looking really near 0. 00:20:14.220 --> 00:20:18.030 And that f of x is really going to be delta f. 00:20:21.110 --> 00:20:25.030 And that g of x is really going to be delta g, because 00:20:25.030 --> 00:20:30.460 let me draw the picture, delta f is that height. 00:20:30.460 --> 00:20:32.450 Here is delta x, and here is the height. 00:20:32.450 --> 00:20:35.790 It's because that point is 0, 0. 00:20:35.790 --> 00:20:45.530 So the differences I'm taking, the f of x in the delta, the f 00:20:45.530 --> 00:20:50.530 of x plus delta x is just f at delta x, just that height. 00:20:50.530 --> 00:20:51.970 And g is this smaller one. 00:20:55.660 --> 00:20:59.150 Do you have an idea of what this answer's going to be? 00:21:02.460 --> 00:21:05.020 If I look at that ratio of this 00:21:05.020 --> 00:21:06.630 function to this function-- 00:21:06.630 --> 00:21:09.740 here the ratio, I don't know what, 3 or something. 00:21:09.740 --> 00:21:13.920 Here it's, I don't know, maybe 4, maybe more. 00:21:13.920 --> 00:21:21.120 As I'm getting closer and closer, this height is 00:21:21.120 --> 00:21:23.690 controlled by the slope. 00:21:23.690 --> 00:21:27.770 And this height, the g of x, is controlled by its slope. 00:21:27.770 --> 00:21:30.060 Look, here is the way to see it. 00:21:30.060 --> 00:21:35.620 Just divide top and bottom by delta x. 00:21:35.620 --> 00:21:37.120 Same thing. 00:21:37.120 --> 00:21:41.140 So I haven't changed anything yet. 00:21:41.140 --> 00:21:45.480 I divided the top and the bottom by delta x, just 00:21:45.480 --> 00:21:53.010 because now I'll let everything go to 0, delta x 00:21:53.010 --> 00:21:56.040 will go to 0, the delta f will go to 0, so the delta 00:21:56.040 --> 00:21:57.130 g will go to 0. 00:21:57.130 --> 00:22:01.200 But I know what this approaches. 00:22:01.200 --> 00:22:08.490 Delta f over delta x approaches the slope, s. 00:22:08.490 --> 00:22:15.190 And delta g over delta x approaches the other slope, t. 00:22:15.190 --> 00:22:22.670 So you see, this is L'Hopital's rule, that if f 00:22:22.670 --> 00:22:29.280 goes to 0, and if g goes to 0, and if they have nice slopes, 00:22:29.280 --> 00:22:34.910 then the ratio of f to g, which looks like 0 over 0, we 00:22:34.910 --> 00:22:39.280 can actually tell what it is by looking at the derivative, 00:22:39.280 --> 00:22:40.970 by looking at those slopes. 00:22:40.970 --> 00:22:44.520 It's the ratio of the slopes. 00:22:44.520 --> 00:22:46.470 OK, that takes a little thought and, of 00:22:46.470 --> 00:22:48.920 course, some practices. 00:22:48.920 --> 00:22:54.400 It also takes some examples to show what else could happen. 00:22:54.400 --> 00:22:59.320 Can I just draw another f, and you tell me what 00:22:59.320 --> 00:23:00.900 about f over g. 00:23:00.900 --> 00:23:03.970 I'm sorry to give you all these questions, but it's 00:23:03.970 --> 00:23:07.830 example, answer, that you get the hang of slopes. 00:23:07.830 --> 00:23:10.670 Suppose f goes much steeper. 00:23:10.670 --> 00:23:13.580 I mean, f could be the square root of x. 00:23:13.580 --> 00:23:16.820 There's f equal the square root of x. 00:23:16.820 --> 00:23:21.260 Square root of x has an infinite slope at 0. 00:23:24.950 --> 00:23:29.340 It's a good function to know, the square root of x, because 00:23:29.340 --> 00:23:32.920 this is x to the 1/2 power. 00:23:32.920 --> 00:23:37.640 And its derivative, its slope, we know will be 1/2 x to the 00:23:37.640 --> 00:23:39.640 minus 1/2 power. 00:23:39.640 --> 00:23:42.270 And then as x goes to 0, that blows up the 00:23:42.270 --> 00:23:45.290 way the picture shows. 00:23:45.290 --> 00:23:53.640 Now, what would f over g, so this is a case where f hasn't 00:23:53.640 --> 00:23:55.460 got a slope. 00:23:55.460 --> 00:23:58.520 The slope is infinite now. s is now infinite. 00:23:58.520 --> 00:24:02.200 And that ratio is going to blow up. 00:24:02.200 --> 00:24:07.443 This one is getting to 0 but slowly. 00:24:11.040 --> 00:24:15.280 This f is staying much bigger than the g, and the ratio 00:24:15.280 --> 00:24:16.190 would be infinite. 00:24:16.190 --> 00:24:21.740 So there's a case where L'Hopital can't help because 00:24:21.740 --> 00:24:26.280 f, this slope s, which was fine for this nice function, 00:24:26.280 --> 00:24:29.000 is not fine for this function. 00:24:29.000 --> 00:24:33.480 The slope is infinite for that square root function. 00:24:33.480 --> 00:24:39.550 OK, a bunch of examples that begin to show what can happen 00:24:39.550 --> 00:24:45.070 and the need, really, for a little bit of care on what 00:24:45.070 --> 00:24:46.460 does it mean? 00:24:46.460 --> 00:24:49.650 What would I say about that square root function? 00:24:53.500 --> 00:24:56.220 So I'll even write that down here. 00:24:56.220 --> 00:25:00.970 f of x equals square root of x at x equals 0. 00:25:00.970 --> 00:25:04.880 What would I say about that function that 00:25:04.880 --> 00:25:06.800 we know it's picture? 00:25:06.800 --> 00:25:10.120 I would say it has infinite slope. 00:25:10.120 --> 00:25:16.330 Or if you prefer, its slope is not defined. 00:25:16.330 --> 00:25:19.090 We don't have a good number there for its slope. 00:25:19.090 --> 00:25:27.350 But I would still say the function is continuous because 00:25:27.350 --> 00:25:34.440 the darn thing does get below any band. 00:25:34.440 --> 00:25:38.240 If I draw a little band here, the function does get into 00:25:38.240 --> 00:25:41.080 that band and stay inside. 00:25:41.080 --> 00:25:43.110 It just took a long time. 00:25:43.110 --> 00:25:45.700 It stayed out of that band as long as it could and then 00:25:45.700 --> 00:25:48.560 finally fell in just at the last minute. 00:25:48.560 --> 00:25:53.910 OK, so I would say this function has the slope not 00:25:53.910 --> 00:25:59.960 defined, not OK at x equals 0. 00:25:59.960 --> 00:26:13.510 But f of x is continuous at x equals 0. 00:26:13.510 --> 00:26:23.920 So I'm trying to make the distinction between asking for 00:26:23.920 --> 00:26:30.110 the function to be continuous is not asking as much. 00:26:30.110 --> 00:26:35.410 If a function's got a nice slope, like g, that function's 00:26:35.410 --> 00:26:36.590 got to be continuous. 00:26:36.590 --> 00:26:39.130 And more, it has to have this good slope. 00:26:39.130 --> 00:26:43.760 This f of x, this square root function will be continuous. 00:26:43.760 --> 00:26:46.590 And now I have to tell you what continuous means. 00:26:46.590 --> 00:26:51.150 It's not asking for so much as a slope, because the slope 00:26:51.150 --> 00:26:56.850 could come down infinitely at the last minute. 00:26:56.850 --> 00:27:00.660 All right, so what's a continuous function? 00:27:00.660 --> 00:27:02.260 Continuous function means-- 00:27:07.550 --> 00:27:13.560 a continuous function, f of x, at some point-- maybe here it 00:27:13.560 --> 00:27:18.190 was 0, I'd better allow any old point. 00:27:18.190 --> 00:27:32.890 So in words, it means f of x approaches f of a as x 00:27:32.890 --> 00:27:36.330 approaches a. 00:27:36.330 --> 00:27:39.430 That's what it means to be continuous at that point. 00:27:39.430 --> 00:27:42.890 It means that there is a number, a value 00:27:42.890 --> 00:27:44.520 for f at that point. 00:27:44.520 --> 00:27:49.600 And we approach that value as we get near that point. 00:27:49.600 --> 00:27:52.040 That seems such a natural idea. 00:27:52.040 --> 00:27:54.850 That's what it means for a function to be continuous. 00:27:54.850 --> 00:28:00.080 And with this piece of chalk or with your pen, it means 00:28:00.080 --> 00:28:04.650 that I can draw the function without lifting my pen. 00:28:04.650 --> 00:28:10.060 Of course, it could do some weird stuff. 00:28:10.060 --> 00:28:11.860 OK, let me just draw here. 00:28:11.860 --> 00:28:16.860 So here's a point, a, and here is my function, f, and 00:28:16.860 --> 00:28:18.150 there is f of a. 00:28:18.150 --> 00:28:21.540 So I'm saying that the function could come along, it 00:28:21.540 --> 00:28:25.350 could come down pretty steeply, but it will get to 00:28:25.350 --> 00:28:26.070 that point. 00:28:26.070 --> 00:28:30.070 It might go on, steeper below, or it might turn back. 00:28:30.070 --> 00:28:31.860 Or it might be level. 00:28:31.860 --> 00:28:39.330 But I can draw the whole thing continuously. 00:28:39.330 --> 00:28:44.290 But now that description with a piece of 00:28:44.290 --> 00:28:47.410 chalk isn't quite enough. 00:28:47.410 --> 00:28:51.870 And there's a formal definition 00:28:51.870 --> 00:28:54.220 that I have to explain. 00:28:54.220 --> 00:28:59.050 And it involves this same idea of epsilon, this 00:28:59.050 --> 00:29:01.520 same idea of a strip. 00:29:01.520 --> 00:29:10.080 It means that if I take a little strip around f of a-- 00:29:10.080 --> 00:29:17.340 so here's f of a plus a little bit, and here's f of a minus a 00:29:17.340 --> 00:29:19.730 little bit-- 00:29:19.730 --> 00:29:21.200 then that's continuous. 00:29:21.200 --> 00:29:23.110 That function is continuous, because-- 00:29:23.110 --> 00:29:27.110 now, remember, epsilon could be smaller than I drew it, 00:29:27.110 --> 00:29:30.810 smaller than I can draw it, but still positive. 00:29:30.810 --> 00:29:35.390 Then the requirement is that it has to get near a, it has 00:29:35.390 --> 00:29:37.560 to get inside that band and stay there. 00:29:37.560 --> 00:29:41.620 It can bounce all over the place. 00:29:41.620 --> 00:29:47.480 But near the point, it's got to get close. 00:29:47.480 --> 00:29:53.780 And now, how do I express that in terms of epsilon? 00:29:53.780 --> 00:29:58.960 OK, well, there's a famous description. 00:29:58.960 --> 00:30:02.685 Yeah, what do I mean by get in there and stay in there? 00:30:06.060 --> 00:30:06.330 Ah! 00:30:06.330 --> 00:30:08.950 Can I just make a story? 00:30:08.950 --> 00:30:12.850 I'm going to use two Greek letters, epsilon and delta, 00:30:12.850 --> 00:30:16.290 hated by all calculus students and professors 00:30:16.290 --> 00:30:17.780 too, if they're truthful. 00:30:17.780 --> 00:30:23.750 OK, so the story goes, we choose a band. 00:30:23.750 --> 00:30:34.060 Ah, since their Greek letters, Socrates chooses epsilon. 00:30:34.060 --> 00:30:35.580 OK. 00:30:35.580 --> 00:30:37.160 So he's going to make it hard. 00:30:37.160 --> 00:30:40.580 He's going to make a narrow band there. 00:30:40.580 --> 00:30:44.380 And then the function has got to get into that band and stay 00:30:44.380 --> 00:30:47.790 there, close to a. 00:30:47.790 --> 00:30:51.530 OK, so what do I mean by close to a? 00:30:51.530 --> 00:30:54.430 Well, that's where delta comes in. 00:30:54.430 --> 00:30:58.920 That's, let's say, Socrates's student, Plato. 00:30:58.920 --> 00:31:10.670 Then Plato can pick his number, delta, which will be 00:31:10.670 --> 00:31:12.730 the width-- 00:31:12.730 --> 00:31:19.540 see, he says, OK, if you get really close, I've got you. 00:31:19.540 --> 00:31:23.610 So he's trying to please Socrates. 00:31:23.610 --> 00:31:29.720 So he says, woo, sorry, a had better be in there somewhere. 00:31:29.720 --> 00:31:34.670 Now these bands are getting so close, my a is, of course-- 00:31:34.670 --> 00:31:40.840 this is really a plus delta, and this guy is a minus delta. 00:31:40.840 --> 00:31:42.090 Are you kind of with me? 00:31:44.780 --> 00:32:02.720 The logic goes, for any epsilon chosen by Socrates, 00:32:02.720 --> 00:32:11.950 Plato can find a positive delta-- 00:32:11.950 --> 00:32:15.030 epsilon, of course, was some positive number, delta might 00:32:15.030 --> 00:32:18.100 be an extremely small positive number-- 00:32:18.100 --> 00:32:27.400 so that if the distance to a is smaller 00:32:27.400 --> 00:32:30.450 than Plato's distance-- 00:32:30.450 --> 00:32:32.750 so if we're in that vertical band-- 00:32:32.750 --> 00:32:36.430 then we're in Socrates's horizontal band. 00:32:36.430 --> 00:32:44.770 Then this f of x minus f of a is below epsilon. 00:32:44.770 --> 00:32:54.730 So Socrates sets up any tough requirement, any horizontal 00:32:54.730 --> 00:32:58.670 band, and then Plato meets that requirement, if the 00:32:58.670 --> 00:33:05.980 function is continuous, by choosing a vertical band that 00:33:05.980 --> 00:33:08.480 keeps everything inside Socrates's band. 00:33:08.480 --> 00:33:09.960 Do you see that? 00:33:09.960 --> 00:33:11.960 Well, it takes some thought. 00:33:11.960 --> 00:33:17.210 It takes some practice, and as always, it's not usually very 00:33:17.210 --> 00:33:19.940 hard to tell if a function is continuous. 00:33:19.940 --> 00:33:22.120 Let me show you one that isn't. 00:33:22.120 --> 00:33:24.340 A famous function that is not continuous. 00:33:27.090 --> 00:33:34.390 Here's the sine of 1 over x as x going to 0. 00:33:34.390 --> 00:33:39.650 What happens to the sine of 1 over x when x goes to 0? 00:33:39.650 --> 00:33:43.620 Well, the sine, we know, oscillates minus 1, plus 1, 00:33:43.620 --> 00:33:45.270 minus 1, plus 1. 00:33:45.270 --> 00:33:51.150 But when it's a sine of 1 over x, that oscillation really 00:33:51.150 --> 00:33:56.730 takes off, because if x gets small, 1 over x is quickly 00:33:56.730 --> 00:33:57.500 getting larger. 00:33:57.500 --> 00:34:02.340 You're running along the sine curve in a faster and faster 00:34:02.340 --> 00:34:03.120 and faster way. 00:34:03.120 --> 00:34:04.050 I can't draw it. 00:34:04.050 --> 00:34:05.570 Here's 0. 00:34:05.570 --> 00:34:12.230 But it's not staying inside a band. 00:34:12.230 --> 00:34:17.239 Even with epsilon equalling 1/2, Socrates has got Plato. 00:34:17.239 --> 00:34:23.900 Plato can't keep it in a band of 1/2 up and 1/2 down because 00:34:23.900 --> 00:34:25.159 the sine doesn't stay there. 00:34:25.159 --> 00:34:27.679 So there's a function that's not continuous. 00:34:27.679 --> 00:34:31.320 I could make it continuous by changing the function a 00:34:31.320 --> 00:34:34.080 little, maybe x times sine of 1 over x. 00:34:34.080 --> 00:34:40.480 That would bring the oscillations down and work. 00:34:40.480 --> 00:34:42.650 So there you go. 00:34:42.650 --> 00:34:44.690 That's epsilon and delta. 00:34:44.690 --> 00:34:46.790 And it takes a little practice. 00:34:46.790 --> 00:34:49.980 And I just have to remember-- 00:34:49.980 --> 00:34:58.500 when you feel that the whole thing is a bad experience-- 00:34:58.500 --> 00:35:01.360 some pity for a Socrates, who actually took poison. 00:35:16.880 --> 00:35:20.680 Not because Plato gave him one that he couldn't do, for some 00:35:20.680 --> 00:35:22.130 completely different reason. 00:35:22.130 --> 00:35:31.350 But this is the meaning of a continuous function, and by 00:35:31.350 --> 00:35:36.880 getting that meaning which took hundreds of years to see. 00:35:36.880 --> 00:35:43.190 And it takes some time to get these two different things, to 00:35:43.190 --> 00:35:44.690 get the logic straight. 00:35:44.690 --> 00:35:50.010 If x is close to a, then f of x is close to f of a. 00:35:50.010 --> 00:35:57.640 That's what this means, f of x approaching f of a. 00:35:57.640 --> 00:36:05.170 That's what Socrates and Plato together had to explain. 00:36:05.170 --> 00:36:08.410 OK, thank you. 00:36:08.410 --> 00:36:10.220 ANNOUNCER: This has been a production of MIT 00:36:10.220 --> 00:36:12.600 OpenCourseWare and Gilbert Strang. 00:36:12.600 --> 00:36:14.880 Funding for this video was provided by the Lord 00:36:14.880 --> 00:36:16.100 Foundation. 00:36:16.100 --> 00:36:19.230 To help OCW continue to provide free and open access 00:36:19.230 --> 00:36:22.310 to MIT courses, please make a donation at 00:36:22.310 --> 00:36:23.870 ocw.mit.edu/donate.