1 00:00:07,580 --> 00:00:08,060 PROFESSOR: OK. 2 00:00:08,060 --> 00:00:12,480 I promised a video about limits and continuous 3 00:00:12,480 --> 00:00:16,370 functions and here it is. 4 00:00:16,370 --> 00:00:23,530 So I'll begin with the most basic idea and with a picture 5 00:00:23,530 --> 00:00:28,730 instead of definition in symbols first. So the most 6 00:00:28,730 --> 00:00:31,710 basic idea is that I have a bunch of numbers-- 7 00:00:31,710 --> 00:00:34,330 let's make them positive numbers-- 8 00:00:34,330 --> 00:00:37,440 and I want to know, what does it mean for them 9 00:00:37,440 --> 00:00:40,220 to approach a limit-- 10 00:00:40,220 --> 00:00:41,580 capital A-- 11 00:00:41,580 --> 00:00:46,810 as I go out this sequence of numbers, a1, a2, a3, a4. 12 00:00:46,810 --> 00:00:50,100 And let me say right away, the first four numbers, the first 13 00:00:50,100 --> 00:00:54,320 million numbers, make no difference about the limit. 14 00:00:54,320 --> 00:00:57,080 So here's what it means. 15 00:00:57,080 --> 00:01:00,970 For example, there's a equals 7, let's say. 16 00:01:00,970 --> 00:01:05,190 What does it mean for these numbers to approach 7? 17 00:01:05,190 --> 00:01:13,840 It means that if I take any thin little space around a, 18 00:01:13,840 --> 00:01:18,450 above and below, the numbers can start out whatever. 19 00:01:18,450 --> 00:01:21,170 They can go in there, they could go out, they could come 20 00:01:21,170 --> 00:01:25,880 back, whatever, they could grow way big, way small. 21 00:01:25,880 --> 00:01:32,700 But in the end, beyond some point, eventually, they have 22 00:01:32,700 --> 00:01:38,310 to get in that slit and stay in there. 23 00:01:38,310 --> 00:01:41,680 And the slit, then, could be smaller. 24 00:01:41,680 --> 00:01:45,220 And then they would have to get into that smaller slit and 25 00:01:45,220 --> 00:01:46,470 stay there. 26 00:01:46,470 --> 00:01:50,510 So that's what it means for the numbers to approach A, 27 00:01:50,510 --> 00:01:56,660 that eventually after any number of jogs around, they 28 00:01:56,660 --> 00:01:59,660 get inside and they stay there, however 29 00:01:59,660 --> 00:02:01,370 thin that slit is. 30 00:02:01,370 --> 00:02:04,180 A slight difference when a is 0, because 31 00:02:04,180 --> 00:02:06,870 the numbers are positive. 32 00:02:06,870 --> 00:02:10,110 They're coming down, they get in. 33 00:02:10,110 --> 00:02:12,920 And again, they must stay in. 34 00:02:12,920 --> 00:02:18,650 And then, again, I'm going to make the band tighter, and 35 00:02:18,650 --> 00:02:20,950 they have to get into that and stay there. 36 00:02:20,950 --> 00:02:25,020 And what does it mean for the numbers to approach infinity? 37 00:02:25,020 --> 00:02:27,270 That means that whatever-- 38 00:02:27,270 --> 00:02:31,080 so this is often called epsilon. 39 00:02:31,080 --> 00:02:34,820 I'll use that Greek letter epsilon as 40 00:02:34,820 --> 00:02:36,290 a very small number. 41 00:02:36,290 --> 00:02:40,100 So this would be A minus epsilon, and this would be A 42 00:02:40,100 --> 00:02:41,500 plus epsilon. 43 00:02:41,500 --> 00:02:44,010 And then the epsilon could be made smaller. 44 00:02:44,010 --> 00:02:45,770 And now here is some big number, 45 00:02:45,770 --> 00:02:48,800 like even 1 over epsilon. 46 00:02:48,800 --> 00:02:50,890 So that's a giant number. 47 00:02:50,890 --> 00:02:55,810 And the limit is infinity if, again, they can dodge around 48 00:02:55,810 --> 00:02:58,060 for a while, they can go down, they can go up. 49 00:02:58,060 --> 00:03:03,160 But eventually, they must get above that 50 00:03:03,160 --> 00:03:05,230 line and stay there. 51 00:03:05,230 --> 00:03:08,190 And if I move the line up further, they have to get 52 00:03:08,190 --> 00:03:11,770 above that line for me to say that the limit is-- 53 00:03:11,770 --> 00:03:15,460 so I have these possible limits. 54 00:03:15,460 --> 00:03:20,180 Infinite, some positive, ordinary number, and 0. 55 00:03:20,180 --> 00:03:22,630 Those are possible limits. 56 00:03:22,630 --> 00:03:27,570 But of course many sequences have no limit at all, like 57 00:03:27,570 --> 00:03:31,280 sine of n, it will just bounce around, cosine n-- 58 00:03:31,280 --> 00:03:32,700 many, many things. 59 00:03:32,700 --> 00:03:33,590 OK. 60 00:03:33,590 --> 00:03:42,280 So I think that the way to get the idea, use the idea, is to 61 00:03:42,280 --> 00:03:46,880 ask some questions about limits. 62 00:03:46,880 --> 00:03:52,570 And we'll see that usually the answer is yes, OK, no problem. 63 00:03:52,570 --> 00:03:57,900 But once in a while, for certain limits are dangerous. 64 00:03:57,900 --> 00:04:03,250 So really always mathematicians are looking for 65 00:04:03,250 --> 00:04:07,370 what's special, what unusual thing could happen? 66 00:04:07,370 --> 00:04:12,980 Because the truth is, limits are ordinarily rather boring. 67 00:04:12,980 --> 00:04:19,220 If the an's approach 7 and the bn's approach 4, so the a's 68 00:04:19,220 --> 00:04:22,220 get close to 7 and the b's get close to 4, then their 69 00:04:22,220 --> 00:04:26,020 differences will get close to 7 minus 4. 70 00:04:28,780 --> 00:04:32,930 But is there any case in which that could fail? 71 00:04:32,930 --> 00:04:38,940 Is there any case among these in which we could not know 72 00:04:38,940 --> 00:04:42,140 what the limit was, and it might not exist, or it might 73 00:04:42,140 --> 00:04:47,040 be like any number? 74 00:04:47,040 --> 00:04:50,400 And I think that can happen in this, so I've got four 75 00:04:50,400 --> 00:04:55,570 different questions here, getting more 76 00:04:55,570 --> 00:04:57,430 interesting as we go down. 77 00:04:57,430 --> 00:05:01,350 In the first one, I can see only one problem. 78 00:05:01,350 --> 00:05:06,770 If the a's approach infinity, so they get very big, and the 79 00:05:06,770 --> 00:05:11,860 b's also approach infinity, get very big, so capital A and 80 00:05:11,860 --> 00:05:21,010 capital B become formally infinity minus infinity, and 81 00:05:21,010 --> 00:05:23,180 we don't know the answer there. 82 00:05:23,180 --> 00:05:24,890 That has no meaning. 83 00:05:24,890 --> 00:05:28,780 So this'll be my little list of danger. 84 00:05:32,680 --> 00:05:35,660 I mean, it's not like skydiving, but for a 85 00:05:35,660 --> 00:05:38,260 mathematician this is high risk. 86 00:05:38,260 --> 00:05:40,760 OK, so how could this happen? 87 00:05:40,760 --> 00:05:44,890 Well, the an might be n squared. 88 00:05:44,890 --> 00:05:47,230 And the bn might be only n. 89 00:05:47,230 --> 00:05:49,230 Right? 90 00:05:49,230 --> 00:05:52,320 So they're both going to infinity, n squared and n. 91 00:05:52,320 --> 00:05:53,820 But n squared is going faster. 92 00:05:53,820 --> 00:05:55,590 It's like a race. 93 00:05:55,590 --> 00:05:59,590 n squared will win, and the difference between them will 94 00:05:59,590 --> 00:06:02,670 actually grow faster and faster. 95 00:06:02,670 --> 00:06:05,410 Or they could go to infinity together. 96 00:06:05,410 --> 00:06:10,110 an and bn could both be n, both headed for infinity. 97 00:06:10,110 --> 00:06:14,640 The differences would be 0 all along, n minus n. 98 00:06:14,640 --> 00:06:17,550 So the limit of the difference would be 0 minus 0. 99 00:06:17,550 --> 00:06:21,060 So this could be 0, but it could be infinity, it could be 100 00:06:21,060 --> 00:06:23,410 minus infinity, it could be anything. 101 00:06:23,410 --> 00:06:25,686 Any limit is possible there. 102 00:06:25,686 --> 00:06:29,610 Do you see that there is a case-- 103 00:06:29,610 --> 00:06:32,800 it's sort of a special case, because it only happens when 104 00:06:32,800 --> 00:06:34,840 these limits are infinite-- 105 00:06:34,840 --> 00:06:41,110 but now it's sort of OK to look at each-- 106 00:06:41,110 --> 00:06:43,230 let me look at number two. 107 00:06:43,230 --> 00:06:46,240 How about multiplication? 108 00:06:46,240 --> 00:06:49,850 If I multiply a bunch of numbers that are headed for 7 109 00:06:49,850 --> 00:06:52,540 and a bunch of numbers that are headed for 4, their 110 00:06:52,540 --> 00:06:55,540 product is going to head for 28. 111 00:06:55,540 --> 00:06:58,650 This will be true. 112 00:06:58,650 --> 00:07:01,960 When could it fail? 113 00:07:01,960 --> 00:07:06,410 Well, again, it's going to be extreme cases, because if I 114 00:07:06,410 --> 00:07:09,270 have ordinary numbers for A and B like 7 and 115 00:07:09,270 --> 00:07:11,690 4, there's no doubt. 116 00:07:11,690 --> 00:07:19,440 But look at the extreme case of when A 117 00:07:19,440 --> 00:07:22,685 is 0 and B is infinite. 118 00:07:26,930 --> 00:07:29,690 So the an's are headed for 0. 119 00:07:29,690 --> 00:07:32,600 The bn's are getting bigger and bigger, the an's are 120 00:07:32,600 --> 00:07:37,000 getting small as we go far enough out. 121 00:07:37,000 --> 00:07:37,690 OK. 122 00:07:37,690 --> 00:07:42,490 In that case, well, again it's a race. 123 00:07:42,490 --> 00:07:46,680 The an's might be 1 over n squared, and the 124 00:07:46,680 --> 00:07:49,250 bn's might be n. 125 00:07:49,250 --> 00:07:53,840 So this would be n over n squared, and 126 00:07:53,840 --> 00:07:55,790 that would go to 0. 127 00:07:55,790 --> 00:08:03,960 But if I reverse those I could have n squared times 1 over n. 128 00:08:03,960 --> 00:08:06,780 The product could get bigger, or the product could-- 129 00:08:06,780 --> 00:08:08,480 all possibilities. 130 00:08:08,480 --> 00:08:09,940 All possibilities there. 131 00:08:09,940 --> 00:08:12,250 So I cannot know what that one is. 132 00:08:12,250 --> 00:08:15,080 0 times infinity is meaningless. 133 00:08:15,080 --> 00:08:15,810 OK. 134 00:08:15,810 --> 00:08:17,680 What about number three? 135 00:08:20,360 --> 00:08:24,610 The danger increases as soon as we start dividing. 136 00:08:24,610 --> 00:08:27,080 I made the b's positive, but I don't know if 137 00:08:27,080 --> 00:08:28,650 capital B is positive. 138 00:08:28,650 --> 00:08:29,960 So the danger-- 139 00:08:29,960 --> 00:08:35,110 and, in fact, the most important case for calculus-- 140 00:08:35,110 --> 00:08:38,480 is 0 over 0. 141 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 142 00:08:43,210 --> 00:08:49,430 their ratio goes to, because it depends how fast they go. 143 00:08:49,430 --> 00:08:53,950 If the a's go quickly to 0 and the b's are rather slow 144 00:08:53,950 --> 00:08:55,260 getting there-- 145 00:08:55,260 --> 00:09:00,060 in other words the b's would be a lot bigger than the a's 146 00:09:00,060 --> 00:09:02,050 even though both are going to 0-- 147 00:09:02,050 --> 00:09:04,730 then that fraction would be small. 148 00:09:04,730 --> 00:09:07,750 But if I reverse them, the fraction would be large. 149 00:09:07,750 --> 00:09:11,180 So I think 0 over 0 is a danger. 150 00:09:11,180 --> 00:09:15,290 I think there's another danger here. 151 00:09:15,290 --> 00:09:17,540 Yeah, maybe infinity over infinity. 152 00:09:17,540 --> 00:09:25,810 Again, that's a race that we can't tell, until we know 153 00:09:25,810 --> 00:09:29,220 details about the sequences, who's going to win. 154 00:09:29,220 --> 00:09:32,530 If they an's go off to infinity and the bn's go off 155 00:09:32,530 --> 00:09:35,910 to infinity, ah, a very important case. 156 00:09:35,910 --> 00:09:39,710 The ratio could be 1 all along. 157 00:09:39,710 --> 00:09:43,420 The a's and b's could be the same, headed for infinity. 158 00:09:43,420 --> 00:09:48,380 Or the an's might be squaring the b's and going up faster, 159 00:09:48,380 --> 00:09:51,195 or the square root of the b's and going slower. 160 00:09:54,710 --> 00:09:58,510 So again, infinity over infinity, we can't-- 161 00:09:58,510 --> 00:10:04,250 0 over infinity, if the a's are headed for 0 and the b's 162 00:10:04,250 --> 00:10:08,400 are headed big, then that ratio is going to be small and 163 00:10:08,400 --> 00:10:09,310 head for 0. 164 00:10:09,310 --> 00:10:11,430 0 over infinity, I'm OK with. 165 00:10:11,430 --> 00:10:13,640 Call it 0. 166 00:10:13,640 --> 00:10:15,350 Well, I don't know if that's legal, but 167 00:10:15,350 --> 00:10:16,870 anyway, let me do it. 168 00:10:16,870 --> 00:10:20,060 All right, last one of this kind, just for 169 00:10:20,060 --> 00:10:21,520 practice about limits. 170 00:10:21,520 --> 00:10:24,990 Again, you see what I'm constantly doing is thinking 171 00:10:24,990 --> 00:10:32,010 of examples that simply show that I can't 172 00:10:32,010 --> 00:10:33,370 be sure of the limit. 173 00:10:33,370 --> 00:10:38,600 So here normally I could be sure, if this is headed for 7 174 00:10:38,600 --> 00:10:42,340 to the fourth, that'll be the limit, whatever that is, 49 175 00:10:42,340 --> 00:10:47,510 squared, 2401, or something. 176 00:10:47,510 --> 00:10:49,680 But if-- 177 00:10:49,680 --> 00:10:52,160 now when could it go wrong? 178 00:10:52,160 --> 00:10:53,990 Here's an interesting case. 179 00:10:53,990 --> 00:10:58,420 So this is my list of danger, and I think I'm in danger if 180 00:10:58,420 --> 00:11:01,000 they both go to 0. 181 00:11:01,000 --> 00:11:05,400 0 to the 0-th power, I don't know what that is. 182 00:11:05,400 --> 00:11:09,210 And actually, I don't know all the possibilities here. 183 00:11:09,210 --> 00:11:19,020 I can see one way would be let's suppose the b's were 184 00:11:19,020 --> 00:11:22,340 actually 0, or practically. 185 00:11:22,340 --> 00:11:26,550 Then things to the 0 power are 1. 186 00:11:26,550 --> 00:11:32,180 So I could get the answer 1 here by fixing the b's at 0 187 00:11:32,180 --> 00:11:35,860 and letting these guys, they would all be to the 0 power, 188 00:11:35,860 --> 00:11:37,440 so they would all be 1. 189 00:11:37,440 --> 00:11:38,945 And in the limit, I would have 1. 190 00:11:42,270 --> 00:11:44,440 But I could also do it differently. 191 00:11:44,440 --> 00:11:49,610 I could fix these at 0 and let these guys get smaller. 192 00:11:49,610 --> 00:11:52,220 Then I would have 0 to powers. 193 00:11:52,220 --> 00:11:54,610 And zero to any power is 0. 194 00:11:54,610 --> 00:11:56,400 You see my little problem here? 195 00:11:56,400 --> 00:11:58,190 Let me write my little problem here. 196 00:11:58,190 --> 00:12:03,620 My problem is that a to the 0 power would be 1, but 0 to the 197 00:12:03,620 --> 00:12:09,435 a-th power would be 0, or 0 to the b-th, maybe I should say. 198 00:12:12,870 --> 00:12:16,840 So if I'm in this situation and the a is shrinking to 0, I 199 00:12:16,840 --> 00:12:18,680 still have a limit of 1's. 200 00:12:18,680 --> 00:12:21,660 But if I'm in this situation and the b's are headed for 0, 201 00:12:21,660 --> 00:12:22,740 I have a limit of 0's. 202 00:12:22,740 --> 00:12:26,540 And maybe you could get 1/2, I don't know how. 203 00:12:26,540 --> 00:12:31,370 And you have to allow me-- because I have to finish this 204 00:12:31,370 --> 00:12:34,540 list, and I only have one more to tell you-- 205 00:12:34,540 --> 00:12:39,120 that another case, a very interesting type of calculus 206 00:12:39,120 --> 00:12:44,020 case is the case where the a's go to 1 and 207 00:12:44,020 --> 00:12:45,345 the b's go to infinity. 208 00:12:48,460 --> 00:12:53,640 I don't know if you remember that this actually happened in 209 00:12:53,640 --> 00:13:00,140 the lecture on e, the number that comes in e to the x, the 210 00:13:00,140 --> 00:13:01,670 great number of calculus. 211 00:13:01,670 --> 00:13:03,440 Do you remember that? 212 00:13:03,440 --> 00:13:06,920 So I'm going to talk a little bit about the a's going to 1 213 00:13:06,920 --> 00:13:10,120 and the b's blowing up. 214 00:13:10,120 --> 00:13:12,960 So I'm getting things that are very near 1, but I'm taking 215 00:13:12,960 --> 00:13:14,830 many, many more of them. 216 00:13:14,830 --> 00:13:17,350 And I believe that I can get all kinds of 217 00:13:17,350 --> 00:13:19,710 different limits there. 218 00:13:19,710 --> 00:13:22,180 I believe I can get all kinds of different limits. 219 00:13:22,180 --> 00:13:23,120 Do you just-- 220 00:13:23,120 --> 00:13:26,180 maybe on this next board. 221 00:13:26,180 --> 00:13:32,710 And then I promise to come back to the heart of the 222 00:13:32,710 --> 00:13:36,270 subject of limits and continuous functions. 223 00:13:36,270 --> 00:13:44,920 But I just think that one, the famous case of this one, was 1 224 00:13:44,920 --> 00:13:47,060 plus 1 over n. 225 00:13:47,060 --> 00:13:51,200 That's the a's, and that approaches what limit? 226 00:13:51,200 --> 00:13:52,810 One. 227 00:13:52,810 --> 00:13:56,120 The b's I'm going to take as n. 228 00:13:56,120 --> 00:13:58,460 So the b's are going to infinity. 229 00:13:58,460 --> 00:14:01,240 So I'm discussing this case here. 230 00:14:01,240 --> 00:14:03,150 So that's a case where this goes to 1, 231 00:14:03,150 --> 00:14:04,260 this goes to infinity. 232 00:14:04,260 --> 00:14:07,992 I had an email this week saying, wait a minute, I've 233 00:14:07,992 --> 00:14:10,540 got a little problem here, because I know 1 to the 234 00:14:10,540 --> 00:14:14,060 infinity is e. 235 00:14:14,060 --> 00:14:15,840 1 to the infinity is 3. 236 00:14:15,840 --> 00:14:23,050 Well, that's because it's true that that number approaches e. 237 00:14:23,050 --> 00:14:27,640 That's one of the many remarkable ways to produce the 238 00:14:27,640 --> 00:14:30,640 number e, the 2.7-something. 239 00:14:30,640 --> 00:14:37,210 But that's because the race between this and this was so 240 00:14:37,210 --> 00:14:38,460 evenly balanced. 241 00:14:41,140 --> 00:14:46,360 If I took these closer and closer to 1, like n squared, 242 00:14:46,360 --> 00:14:48,420 what would happen then? 243 00:14:48,420 --> 00:14:52,070 Then I'm taking numbers very, very near 1, I'm taking a 244 00:14:52,070 --> 00:14:55,530 power, but these are sort of near, those would approach-- 245 00:14:55,530 --> 00:14:58,660 would you like to guess? 246 00:14:58,660 --> 00:15:00,820 One. 247 00:15:00,820 --> 00:15:03,990 These are so close to 1 that taking the nth power doesn't 248 00:15:03,990 --> 00:15:05,690 move them far. 249 00:15:05,690 --> 00:15:12,070 And you can guess that I could get infinity too, by taking n 250 00:15:12,070 --> 00:15:14,860 not still close to 1 and taking some big 251 00:15:14,860 --> 00:15:18,950 power like n squared. 252 00:15:18,950 --> 00:15:21,950 Now I have things close to 1, but I'm taking so many of them 253 00:15:21,950 --> 00:15:23,200 that it would blow up. 254 00:15:25,820 --> 00:15:29,120 So again i think--So those are all cases where in the limit, 255 00:15:29,120 --> 00:15:31,780 I have 1, in the limit, I have infinity. 256 00:15:31,780 --> 00:15:41,620 But that combination 1 to the increasingly high powers can 257 00:15:41,620 --> 00:15:43,790 do different things. 258 00:15:43,790 --> 00:15:49,790 This was my little idea to show you the risky cases. 259 00:15:49,790 --> 00:15:50,820 OK. 260 00:15:50,820 --> 00:15:57,960 But actually, 0 over 0, that's what calculus is 261 00:15:57,960 --> 00:16:00,290 always doing, right? 262 00:16:00,290 --> 00:16:04,730 Because that's exactly what we have when we have a delta f 263 00:16:04,730 --> 00:16:08,310 over a delta x, a delta y over a delta x. 264 00:16:08,310 --> 00:16:14,010 They're both approaching 0 and we get a definite slope when 265 00:16:14,010 --> 00:16:17,420 the ratio goes to a good number. 266 00:16:17,420 --> 00:16:18,340 OK. 267 00:16:18,340 --> 00:16:21,980 So can I discuss 0 over 0? 268 00:16:21,980 --> 00:16:25,530 All right, phooey on this one. 269 00:16:25,530 --> 00:16:26,250 OK. 270 00:16:26,250 --> 00:16:32,560 So I now want to speak about the case when f 271 00:16:32,560 --> 00:16:33,810 of x goes to 0. 272 00:16:37,140 --> 00:16:41,080 Let's say f of x goes to 0 as x goes to 0. 273 00:16:41,080 --> 00:16:45,570 So there'll be an if here. 274 00:16:45,570 --> 00:16:48,680 I have to say what that means. 275 00:16:48,680 --> 00:16:55,430 And then I'm also going to have some g of x going to 0 as 276 00:16:55,430 --> 00:16:57,760 x goes to 0. 277 00:16:57,760 --> 00:17:02,210 OK, so both functions are decreasing. 278 00:17:02,210 --> 00:17:07,530 And my question, let me ask the question first, what about 279 00:17:07,530 --> 00:17:09,740 f of x over g of x? 280 00:17:12,339 --> 00:17:14,150 What does that do? 281 00:17:14,150 --> 00:17:22,499 And of course, just as I said up there, I can't tell yet. 282 00:17:25,420 --> 00:17:27,510 I have to know the f and the g. 283 00:17:27,510 --> 00:17:31,190 It's a race to 0, and I have to know who's the winner and 284 00:17:31,190 --> 00:17:32,710 by how much. 285 00:17:32,710 --> 00:17:36,450 But first, I'd better say what does it mean for a function to 286 00:17:36,450 --> 00:17:38,900 go to 0 as x goes to 0. 287 00:17:38,900 --> 00:17:40,230 Well, you know. 288 00:17:40,230 --> 00:17:44,140 Let me draw a graph of this function. 289 00:17:44,140 --> 00:17:45,690 OK, I'll just draw it. 290 00:17:45,690 --> 00:17:47,390 So f of x is going to 0. 291 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. 292 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. 293 00:18:00,850 --> 00:18:05,350 So it could come like so. 294 00:18:05,350 --> 00:18:10,460 That's a pretty sensible, smooth, nice approach to 0. 295 00:18:10,460 --> 00:18:14,300 That could be my f of x. 296 00:18:14,300 --> 00:18:21,400 And it may be a g of x is smaller, but also approaching 297 00:18:21,400 --> 00:18:23,590 0 in a nice, smooth way. 298 00:18:23,590 --> 00:18:29,720 This is a case where you can see those, 299 00:18:29,720 --> 00:18:32,020 as x goes that way-- 300 00:18:32,020 --> 00:18:34,680 maybe the arrow should be going that way, because x is 301 00:18:34,680 --> 00:18:36,590 going to 0-- 302 00:18:36,590 --> 00:18:41,690 my f of x is getting smaller, my g of x is getting smaller. 303 00:18:41,690 --> 00:18:46,090 And I'll say exactly what that means, but you 304 00:18:46,090 --> 00:18:47,320 know what it means. 305 00:18:47,320 --> 00:18:51,750 It means that if I put a little, like these lines, if I 306 00:18:51,750 --> 00:18:56,160 put a little band there, it gets into that band. 307 00:18:56,160 --> 00:18:59,610 Actually, g will get into the band sooner. 308 00:18:59,610 --> 00:19:03,300 But then f will safely get into the band. 309 00:19:03,300 --> 00:19:08,760 Now, the question is what about f of x over g of x? 310 00:19:08,760 --> 00:19:10,510 OK. 311 00:19:10,510 --> 00:19:11,660 Can we say? 312 00:19:11,660 --> 00:19:16,840 Now, I'm going to suppose that f of x has a 313 00:19:16,840 --> 00:19:20,100 definite slope, s. 314 00:19:20,100 --> 00:19:22,975 And this one has a definite slope, t. 315 00:19:26,750 --> 00:19:26,998 In other words, I am going to suppose-- 316 00:19:26,998 --> 00:19:32,950 Here look, this is called, named after a French guy, 317 00:19:32,950 --> 00:19:38,470 L'Hopital, the hospital rule. 318 00:19:38,470 --> 00:19:49,320 OK, so it's just a little trick, because this comes up 319 00:19:49,320 --> 00:19:54,320 of what's happening in this race to 0. 320 00:19:54,320 --> 00:20:01,280 And the natural idea is that f of x is really, since f is 0 321 00:20:01,280 --> 00:20:05,820 there, and I'm really just going a little way. 322 00:20:05,820 --> 00:20:11,170 So maybe I call that delta x, just to emphasize that I'm 323 00:20:11,170 --> 00:20:14,220 looking really near 0. 324 00:20:14,220 --> 00:20:18,030 And that f of x is really going to be delta f. 325 00:20:21,110 --> 00:20:25,030 And that g of x is really going to be delta g, because 326 00:20:25,030 --> 00:20:30,460 let me draw the picture, delta f is that height. 327 00:20:30,460 --> 00:20:32,450 Here is delta x, and here is the height. 328 00:20:32,450 --> 00:20:35,790 It's because that point is 0, 0. 329 00:20:35,790 --> 00:20:45,530 So the differences I'm taking, the f of x in the delta, the f 330 00:20:45,530 --> 00:20:50,530 of x plus delta x is just f at delta x, just that height. 331 00:20:50,530 --> 00:20:51,970 And g is this smaller one. 332 00:20:55,660 --> 00:20:59,150 Do you have an idea of what this answer's going to be? 333 00:21:02,460 --> 00:21:05,020 If I look at that ratio of this 334 00:21:05,020 --> 00:21:06,630 function to this function-- 335 00:21:06,630 --> 00:21:09,740 here the ratio, I don't know what, 3 or something. 336 00:21:09,740 --> 00:21:13,920 Here it's, I don't know, maybe 4, maybe more. 337 00:21:13,920 --> 00:21:21,120 As I'm getting closer and closer, this height is 338 00:21:21,120 --> 00:21:23,690 controlled by the slope. 339 00:21:23,690 --> 00:21:27,770 And this height, the g of x, is controlled by its slope. 340 00:21:27,770 --> 00:21:30,060 Look, here is the way to see it. 341 00:21:30,060 --> 00:21:35,620 Just divide top and bottom by delta x. 342 00:21:35,620 --> 00:21:37,120 Same thing. 343 00:21:37,120 --> 00:21:41,140 So I haven't changed anything yet. 344 00:21:41,140 --> 00:21:45,480 I divided the top and the bottom by delta x, just 345 00:21:45,480 --> 00:21:53,010 because now I'll let everything go to 0, delta x 346 00:21:53,010 --> 00:21:56,040 will go to 0, the delta f will go to 0, so the delta 347 00:21:56,040 --> 00:21:57,130 g will go to 0. 348 00:21:57,130 --> 00:22:01,200 But I know what this approaches. 349 00:22:01,200 --> 00:22:08,490 Delta f over delta x approaches the slope, s. 350 00:22:08,490 --> 00:22:15,190 And delta g over delta x approaches the other slope, t. 351 00:22:15,190 --> 00:22:22,670 So you see, this is L'Hopital's rule, that if f 352 00:22:22,670 --> 00:22:29,280 goes to 0, and if g goes to 0, and if they have nice slopes, 353 00:22:29,280 --> 00:22:34,910 then the ratio of f to g, which looks like 0 over 0, we 354 00:22:34,910 --> 00:22:39,280 can actually tell what it is by looking at the derivative, 355 00:22:39,280 --> 00:22:40,970 by looking at those slopes. 356 00:22:40,970 --> 00:22:44,520 It's the ratio of the slopes. 357 00:22:44,520 --> 00:22:46,470 OK, that takes a little thought and, of 358 00:22:46,470 --> 00:22:48,920 course, some practices. 359 00:22:48,920 --> 00:22:54,400 It also takes some examples to show what else could happen. 360 00:22:54,400 --> 00:22:59,320 Can I just draw another f, and you tell me what 361 00:22:59,320 --> 00:23:00,900 about f over g. 362 00:23:00,900 --> 00:23:03,970 I'm sorry to give you all these questions, but it's 363 00:23:03,970 --> 00:23:07,830 example, answer, that you get the hang of slopes. 364 00:23:07,830 --> 00:23:10,670 Suppose f goes much steeper. 365 00:23:10,670 --> 00:23:13,580 I mean, f could be the square root of x. 366 00:23:13,580 --> 00:23:16,820 There's f equal the square root of x. 367 00:23:16,820 --> 00:23:21,260 Square root of x has an infinite slope at 0. 368 00:23:24,950 --> 00:23:29,340 It's a good function to know, the square root of x, because 369 00:23:29,340 --> 00:23:32,920 this is x to the 1/2 power. 370 00:23:32,920 --> 00:23:37,640 And its derivative, its slope, we know will be 1/2 x to the 371 00:23:37,640 --> 00:23:39,640 minus 1/2 power. 372 00:23:39,640 --> 00:23:42,270 And then as x goes to 0, that blows up the 373 00:23:42,270 --> 00:23:45,290 way the picture shows. 374 00:23:45,290 --> 00:23:53,640 Now, what would f over g, so this is a case where f hasn't 375 00:23:53,640 --> 00:23:55,460 got a slope. 376 00:23:55,460 --> 00:23:58,520 The slope is infinite now. s is now infinite. 377 00:23:58,520 --> 00:24:02,200 And that ratio is going to blow up. 378 00:24:02,200 --> 00:24:07,443 This one is getting to 0 but slowly. 379 00:24:11,040 --> 00:24:15,280 This f is staying much bigger than the g, and the ratio 380 00:24:15,280 --> 00:24:16,190 would be infinite. 381 00:24:16,190 --> 00:24:21,740 So there's a case where L'Hopital can't help because 382 00:24:21,740 --> 00:24:26,280 f, this slope s, which was fine for this nice function, 383 00:24:26,280 --> 00:24:29,000 is not fine for this function. 384 00:24:29,000 --> 00:24:33,480 The slope is infinite for that square root function. 385 00:24:33,480 --> 00:24:39,550 OK, a bunch of examples that begin to show what can happen 386 00:24:39,550 --> 00:24:45,070 and the need, really, for a little bit of care on what 387 00:24:45,070 --> 00:24:46,460 does it mean? 388 00:24:46,460 --> 00:24:49,650 What would I say about that square root function? 389 00:24:53,500 --> 00:24:56,220 So I'll even write that down here. 390 00:24:56,220 --> 00:25:00,970 f of x equals square root of x at x equals 0. 391 00:25:00,970 --> 00:25:04,880 What would I say about that function that 392 00:25:04,880 --> 00:25:06,800 we know it's picture? 393 00:25:06,800 --> 00:25:10,120 I would say it has infinite slope. 394 00:25:10,120 --> 00:25:16,330 Or if you prefer, its slope is not defined. 395 00:25:16,330 --> 00:25:19,090 We don't have a good number there for its slope. 396 00:25:19,090 --> 00:25:27,350 But I would still say the function is continuous because 397 00:25:27,350 --> 00:25:34,440 the darn thing does get below any band. 398 00:25:34,440 --> 00:25:38,240 If I draw a little band here, the function does get into 399 00:25:38,240 --> 00:25:41,080 that band and stay inside. 400 00:25:41,080 --> 00:25:43,110 It just took a long time. 401 00:25:43,110 --> 00:25:45,700 It stayed out of that band as long as it could and then 402 00:25:45,700 --> 00:25:48,560 finally fell in just at the last minute. 403 00:25:48,560 --> 00:25:53,910 OK, so I would say this function has the slope not 404 00:25:53,910 --> 00:25:59,960 defined, not OK at x equals 0. 405 00:25:59,960 --> 00:26:13,510 But f of x is continuous at x equals 0. 406 00:26:13,510 --> 00:26:23,920 So I'm trying to make the distinction between asking for 407 00:26:23,920 --> 00:26:30,110 the function to be continuous is not asking as much. 408 00:26:30,110 --> 00:26:35,410 If a function's got a nice slope, like g, that function's 409 00:26:35,410 --> 00:26:36,590 got to be continuous. 410 00:26:36,590 --> 00:26:39,130 And more, it has to have this good slope. 411 00:26:39,130 --> 00:26:43,760 This f of x, this square root function will be continuous. 412 00:26:43,760 --> 00:26:46,590 And now I have to tell you what continuous means. 413 00:26:46,590 --> 00:26:51,150 It's not asking for so much as a slope, because the slope 414 00:26:51,150 --> 00:26:56,850 could come down infinitely at the last minute. 415 00:26:56,850 --> 00:27:00,660 All right, so what's a continuous function? 416 00:27:00,660 --> 00:27:02,260 Continuous function means-- 417 00:27:07,550 --> 00:27:13,560 a continuous function, f of x, at some point-- maybe here it 418 00:27:13,560 --> 00:27:18,190 was 0, I'd better allow any old point. 419 00:27:18,190 --> 00:27:32,890 So in words, it means f of x approaches f of a as x 420 00:27:32,890 --> 00:27:36,330 approaches a. 421 00:27:36,330 --> 00:27:39,430 That's what it means to be continuous at that point. 422 00:27:39,430 --> 00:27:42,890 It means that there is a number, a value 423 00:27:42,890 --> 00:27:44,520 for f at that point. 424 00:27:44,520 --> 00:27:49,600 And we approach that value as we get near that point. 425 00:27:49,600 --> 00:27:52,040 That seems such a natural idea. 426 00:27:52,040 --> 00:27:54,850 That's what it means for a function to be continuous. 427 00:27:54,850 --> 00:28:00,080 And with this piece of chalk or with your pen, it means 428 00:28:00,080 --> 00:28:04,650 that I can draw the function without lifting my pen. 429 00:28:04,650 --> 00:28:10,060 Of course, it could do some weird stuff. 430 00:28:10,060 --> 00:28:11,860 OK, let me just draw here. 431 00:28:11,860 --> 00:28:16,860 So here's a point, a, and here is my function, f, and 432 00:28:16,860 --> 00:28:18,150 there is f of a. 433 00:28:18,150 --> 00:28:21,540 So I'm saying that the function could come along, it 434 00:28:21,540 --> 00:28:25,350 could come down pretty steeply, but it will get to 435 00:28:25,350 --> 00:28:26,070 that point. 436 00:28:26,070 --> 00:28:30,070 It might go on, steeper below, or it might turn back. 437 00:28:30,070 --> 00:28:31,860 Or it might be level. 438 00:28:31,860 --> 00:28:39,330 But I can draw the whole thing continuously. 439 00:28:39,330 --> 00:28:44,290 But now that description with a piece of 440 00:28:44,290 --> 00:28:47,410 chalk isn't quite enough. 441 00:28:47,410 --> 00:28:51,870 And there's a formal definition 442 00:28:51,870 --> 00:28:54,220 that I have to explain. 443 00:28:54,220 --> 00:28:59,050 And it involves this same idea of epsilon, this 444 00:28:59,050 --> 00:29:01,520 same idea of a strip. 445 00:29:01,520 --> 00:29:10,080 It means that if I take a little strip around f of a-- 446 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 447 00:29:17,340 --> 00:29:19,730 little bit-- 448 00:29:19,730 --> 00:29:21,200 then that's continuous. 449 00:29:21,200 --> 00:29:23,110 That function is continuous, because-- 450 00:29:23,110 --> 00:29:27,110 now, remember, epsilon could be smaller than I drew it, 451 00:29:27,110 --> 00:29:30,810 smaller than I can draw it, but still positive. 452 00:29:30,810 --> 00:29:35,390 Then the requirement is that it has to get near a, it has 453 00:29:35,390 --> 00:29:37,560 to get inside that band and stay there. 454 00:29:37,560 --> 00:29:41,620 It can bounce all over the place. 455 00:29:41,620 --> 00:29:47,480 But near the point, it's got to get close. 456 00:29:47,480 --> 00:29:53,780 And now, how do I express that in terms of epsilon? 457 00:29:53,780 --> 00:29:58,960 OK, well, there's a famous description. 458 00:29:58,960 --> 00:30:02,685 Yeah, what do I mean by get in there and stay in there? 459 00:30:06,060 --> 00:30:06,330 Ah! 460 00:30:06,330 --> 00:30:08,950 Can I just make a story? 461 00:30:08,950 --> 00:30:12,850 I'm going to use two Greek letters, epsilon and delta, 462 00:30:12,850 --> 00:30:16,290 hated by all calculus students and professors 463 00:30:16,290 --> 00:30:17,780 too, if they're truthful. 464 00:30:17,780 --> 00:30:23,750 OK, so the story goes, we choose a band. 465 00:30:23,750 --> 00:30:34,060 Ah, since their Greek letters, Socrates chooses epsilon. 466 00:30:34,060 --> 00:30:35,580 OK. 467 00:30:35,580 --> 00:30:37,160 So he's going to make it hard. 468 00:30:37,160 --> 00:30:40,580 He's going to make a narrow band there. 469 00:30:40,580 --> 00:30:44,380 And then the function has got to get into that band and stay 470 00:30:44,380 --> 00:30:47,790 there, close to a. 471 00:30:47,790 --> 00:30:51,530 OK, so what do I mean by close to a? 472 00:30:51,530 --> 00:30:54,430 Well, that's where delta comes in. 473 00:30:54,430 --> 00:30:58,920 That's, let's say, Socrates's student, Plato. 474 00:30:58,920 --> 00:31:10,670 Then Plato can pick his number, delta, which will be 475 00:31:10,670 --> 00:31:12,730 the width-- 476 00:31:12,730 --> 00:31:19,540 see, he says, OK, if you get really close, I've got you. 477 00:31:19,540 --> 00:31:23,610 So he's trying to please Socrates. 478 00:31:23,610 --> 00:31:29,720 So he says, woo, sorry, a had better be in there somewhere. 479 00:31:29,720 --> 00:31:34,670 Now these bands are getting so close, my a is, of course-- 480 00:31:34,670 --> 00:31:40,840 this is really a plus delta, and this guy is a minus delta. 481 00:31:40,840 --> 00:31:42,090 Are you kind of with me? 482 00:31:44,780 --> 00:32:02,720 The logic goes, for any epsilon chosen by Socrates, 483 00:32:02,720 --> 00:32:11,950 Plato can find a positive delta-- 484 00:32:11,950 --> 00:32:15,030 epsilon, of course, was some positive number, delta might 485 00:32:15,030 --> 00:32:18,100 be an extremely small positive number-- 486 00:32:18,100 --> 00:32:27,400 so that if the distance to a is smaller 487 00:32:27,400 --> 00:32:30,450 than Plato's distance-- 488 00:32:30,450 --> 00:32:32,750 so if we're in that vertical band-- 489 00:32:32,750 --> 00:32:36,430 then we're in Socrates's horizontal band. 490 00:32:36,430 --> 00:32:44,770 Then this f of x minus f of a is below epsilon. 491 00:32:44,770 --> 00:32:54,730 So Socrates sets up any tough requirement, any horizontal 492 00:32:54,730 --> 00:32:58,670 band, and then Plato meets that requirement, if the 493 00:32:58,670 --> 00:33:05,980 function is continuous, by choosing a vertical band that 494 00:33:05,980 --> 00:33:08,480 keeps everything inside Socrates's band. 495 00:33:08,480 --> 00:33:09,960 Do you see that? 496 00:33:09,960 --> 00:33:11,960 Well, it takes some thought. 497 00:33:11,960 --> 00:33:17,210 It takes some practice, and as always, it's not usually very 498 00:33:17,210 --> 00:33:19,940 hard to tell if a function is continuous. 499 00:33:19,940 --> 00:33:22,120 Let me show you one that isn't. 500 00:33:22,120 --> 00:33:24,340 A famous function that is not continuous. 501 00:33:27,090 --> 00:33:34,390 Here's the sine of 1 over x as x going to 0. 502 00:33:34,390 --> 00:33:39,650 What happens to the sine of 1 over x when x goes to 0? 503 00:33:39,650 --> 00:33:43,620 Well, the sine, we know, oscillates minus 1, plus 1, 504 00:33:43,620 --> 00:33:45,270 minus 1, plus 1. 505 00:33:45,270 --> 00:33:51,150 But when it's a sine of 1 over x, that oscillation really 506 00:33:51,150 --> 00:33:56,730 takes off, because if x gets small, 1 over x is quickly 507 00:33:56,730 --> 00:33:57,500 getting larger. 508 00:33:57,500 --> 00:34:02,340 You're running along the sine curve in a faster and faster 509 00:34:02,340 --> 00:34:03,120 and faster way. 510 00:34:03,120 --> 00:34:04,050 I can't draw it. 511 00:34:04,050 --> 00:34:05,570 Here's 0. 512 00:34:05,570 --> 00:34:12,230 But it's not staying inside a band. 513 00:34:12,230 --> 00:34:17,239 Even with epsilon equalling 1/2, Socrates has got Plato. 514 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 515 00:34:23,900 --> 00:34:25,159 the sine doesn't stay there. 516 00:34:25,159 --> 00:34:27,679 So there's a function that's not continuous. 517 00:34:27,679 --> 00:34:31,320 I could make it continuous by changing the function a 518 00:34:31,320 --> 00:34:34,080 little, maybe x times sine of 1 over x. 519 00:34:34,080 --> 00:34:40,480 That would bring the oscillations down and work. 520 00:34:40,480 --> 00:34:42,650 So there you go. 521 00:34:42,650 --> 00:34:44,690 That's epsilon and delta. 522 00:34:44,690 --> 00:34:46,790 And it takes a little practice. 523 00:34:46,790 --> 00:34:49,980 And I just have to remember-- 524 00:34:49,980 --> 00:34:58,500 when you feel that the whole thing is a bad experience-- 525 00:34:58,500 --> 00:35:01,360 some pity for a Socrates, who actually took poison. 526 00:35:16,880 --> 00:35:20,680 Not because Plato gave him one that he couldn't do, for some 527 00:35:20,680 --> 00:35:22,130 completely different reason. 528 00:35:22,130 --> 00:35:31,350 But this is the meaning of a continuous function, and by 529 00:35:31,350 --> 00:35:36,880 getting that meaning which took hundreds of years to see. 530 00:35:36,880 --> 00:35:43,190 And it takes some time to get these two different things, to 531 00:35:43,190 --> 00:35:44,690 get the logic straight. 532 00:35:44,690 --> 00:35:50,010 If x is close to a, then f of x is close to f of a. 533 00:35:50,010 --> 00:35:57,640 That's what this means, f of x approaching f of a. 534 00:35:57,640 --> 00:36:05,170 That's what Socrates and Plato together had to explain. 535 00:36:05,170 --> 00:36:08,410 OK, thank you. 536 00:36:08,410 --> 00:36:10,220 ANNOUNCER: This has been a production of MIT 537 00:36:10,220 --> 00:36:12,600 OpenCourseWare and Gilbert Strang. 538 00:36:12,600 --> 00:36:14,880 Funding for this video was provided by the Lord 539 00:36:14,880 --> 00:36:16,100 Foundation. 540 00:36:16,100 --> 00:36:19,230 To help OCW continue to provide free and open access 541 00:36:19,230 --> 00:36:22,310 to MIT courses, please make a donation at 542 00:36:22,310 --> 00:36:23,870 ocw.mit.edu/donate.