1 00:00:00,000 --> 00:00:00,900 2 00:00:00,900 --> 00:00:02,810 Let me draw a function that would be interesting 3 00:00:02,810 --> 00:00:04,490 to take a limit of. 4 00:00:04,490 --> 00:00:06,880 And I'll just draw it visually for now, and we'll do some 5 00:00:06,879 --> 00:00:08,390 specific examples a little later. 6 00:00:08,390 --> 00:00:11,870 So that's my y-axis, and that's my x-axis. 7 00:00:11,869 --> 00:00:14,179 And let;s say the function looks something like-- 8 00:00:14,179 --> 00:00:15,949 I'll make it a fairly straightforward function 9 00:00:15,949 --> 00:00:19,759 --let's say it's a line, for the most part. 10 00:00:19,760 --> 00:00:23,100 Let's say it looks just like, accept it has a 11 00:00:23,100 --> 00:00:27,080 hole at some point. 12 00:00:27,079 --> 00:00:28,689 x is equal to a, so it's undefined there. 13 00:00:28,690 --> 00:00:32,030 Let me black that point out so you can see that 14 00:00:32,030 --> 00:00:33,109 it's not defined there. 15 00:00:33,109 --> 00:00:38,780 And that point there is x is equal to a. 16 00:00:38,780 --> 00:00:45,179 This is the x-axis, this is the y is equal f of x-axis. 17 00:00:45,179 --> 00:00:47,119 Let's just say that's the y-axis. 18 00:00:47,119 --> 00:00:51,030 And let's say that this is f of x, or this is 19 00:00:51,030 --> 00:00:53,880 y is equal to f of x. 20 00:00:53,880 --> 00:00:55,740 Now we've done a bunch of videos on limits. 21 00:00:55,740 --> 00:00:57,160 I think you have an intuition on this. 22 00:00:57,159 --> 00:00:59,849 If I were to say what is the limit as x approaches a, 23 00:00:59,850 --> 00:01:04,019 and let's say that this point right here is l. 24 00:01:04,019 --> 00:01:06,479 We know from our previous videos that-- well first of all 25 00:01:06,480 --> 00:01:10,939 I could write it down --the limit as x approaches 26 00:01:10,939 --> 00:01:13,689 a of f of x. 27 00:01:13,689 --> 00:01:17,560 What this means intuitively is as we approach a from either 28 00:01:17,560 --> 00:01:20,980 side, as we approach it from that side, what does 29 00:01:20,980 --> 00:01:22,290 f of x approach? 30 00:01:22,290 --> 00:01:27,030 So when x is here, f of x is here. 31 00:01:27,030 --> 00:01:29,489 When x is here, f of x is there. 32 00:01:29,489 --> 00:01:33,079 And we see that it's approaching this l right there. 33 00:01:33,079 --> 00:01:35,950 34 00:01:35,950 --> 00:01:40,320 And when we approach a from that side-- and we've done 35 00:01:40,319 --> 00:01:42,199 limits where you approach from only the left or right side, 36 00:01:42,200 --> 00:01:44,750 but to actually have a limit it has to approach the same thing 37 00:01:44,750 --> 00:01:48,670 from the positive direction and the negative direction --but as 38 00:01:48,670 --> 00:01:52,379 you go from there, if you pick this x, then this is f of x. 39 00:01:52,379 --> 00:01:54,439 f of x is right there. 40 00:01:54,439 --> 00:01:57,459 If x gets here then it goes here, and as we get closer and 41 00:01:57,459 --> 00:02:03,859 closer to a, f of x approaches this point l, or this value l. 42 00:02:03,859 --> 00:02:06,599 So we say that the limit of f of x ax x approaches 43 00:02:06,599 --> 00:02:07,959 a is equal to l. 44 00:02:07,959 --> 00:02:09,639 I think we have that intuition. 45 00:02:09,639 --> 00:02:13,359 But this was not very, it's actually not rigorous at all 46 00:02:13,360 --> 00:02:15,480 in terms of being specific in terms of what we 47 00:02:15,479 --> 00:02:16,289 mean is a limit. 48 00:02:16,289 --> 00:02:19,340 All I said so far is as we get closer, what does 49 00:02:19,340 --> 00:02:21,439 f of x get closer to? 50 00:02:21,439 --> 00:02:27,359 So in this video I'll attempt to explain to you a definition 51 00:02:27,360 --> 00:02:29,360 of a limit that has a little bit more, or actually a lot 52 00:02:29,360 --> 00:02:32,180 more, mathematical rigor than just saying you know, as x gets 53 00:02:32,180 --> 00:02:36,990 closer to this value, what does f of x get closer to? 54 00:02:36,990 --> 00:02:39,290 And the way I think about it's: kind of like a little game. 55 00:02:39,289 --> 00:02:48,639 The definition is, this statement right here means that 56 00:02:48,639 --> 00:02:55,149 I can always give you a range about this point-- and when I 57 00:02:55,150 --> 00:02:57,189 talk about range I'm not talking about it in the whole 58 00:02:57,189 --> 00:03:00,960 domain range aspect, I'm just talking about a range like you 59 00:03:00,960 --> 00:03:05,980 know, I can give you a distance from a as long as I'm no 60 00:03:05,979 --> 00:03:12,359 further than that, I can guarantee you that f of x is go 61 00:03:12,360 --> 00:03:16,160 it not going to be any further than a given distance from l 62 00:03:16,159 --> 00:03:18,030 --and the way I think about it is, it could be viewed 63 00:03:18,030 --> 00:03:18,490 as a little game. 64 00:03:18,490 --> 00:03:21,840 Let's say you say, OK Sal, I don't believe you. 65 00:03:21,840 --> 00:03:29,900 I want to see you know, whether f of x can get within 0.5 of l. 66 00:03:29,900 --> 00:03:37,460 So let's say you give me 0.5 and you say Sal, by this 67 00:03:37,460 --> 00:03:39,760 definition you should always be able to give me a range 68 00:03:39,759 --> 00:03:46,329 around a that will get f of x within 0.5 of l, right? 69 00:03:46,330 --> 00:03:49,980 So the values of f of x are always going to be right in 70 00:03:49,979 --> 00:03:51,159 this range, right there. 71 00:03:51,159 --> 00:03:54,299 And as long as I'm in that range around a, as long as I'm 72 00:03:54,300 --> 00:03:57,890 the range around you give me, f of x will always be at least 73 00:03:57,889 --> 00:04:00,029 that close to our limit point. 74 00:04:00,030 --> 00:04:02,819 75 00:04:02,819 --> 00:04:07,829 Let me draw it a little bit bigger, just because I think 76 00:04:07,830 --> 00:04:10,870 I'm just overriding the same diagram over and over again. 77 00:04:10,870 --> 00:04:16,769 So let's say that this is f of x, this is the hole point. 78 00:04:16,769 --> 00:04:19,339 There doesn't have to be a hole there; the limit could equal 79 00:04:19,339 --> 00:04:21,019 actually a value of the function, but the limit is more 80 00:04:21,019 --> 00:04:22,560 interesting when the function isn't defined there 81 00:04:22,560 --> 00:04:23,910 but the limit is. 82 00:04:23,910 --> 00:04:28,770 So this point right here-- that is, let me draw the axes again. 83 00:04:28,769 --> 00:04:31,529 84 00:04:31,529 --> 00:04:44,009 So that's x-axis, y-axis x, y, this is the limit point 85 00:04:44,009 --> 00:04:47,310 l, this is the point a. 86 00:04:47,310 --> 00:04:49,629 So the definition of the limit, and I'll go back to this in 87 00:04:49,629 --> 00:04:52,689 second because now that it's bigger I want explain it again. 88 00:04:52,689 --> 00:04:58,089 It says this means-- and this is the epsilon delta definition 89 00:04:58,089 --> 00:05:01,259 of limits, and we'll touch on epsilon and delta in a second, 90 00:05:01,259 --> 00:05:05,789 is I can guarantee you that f of x, you give me any 91 00:05:05,790 --> 00:05:08,860 distance from l you want. 92 00:05:08,860 --> 00:05:10,449 And actually let's call that epsilon. 93 00:05:10,449 --> 00:05:12,589 And let's just hit on the definition right 94 00:05:12,589 --> 00:05:13,049 from the get go. 95 00:05:13,050 --> 00:05:17,090 So you say I want to be no more than epsilon away from l. 96 00:05:17,089 --> 00:05:19,509 And epsilon can just be any number greater, any real 97 00:05:19,509 --> 00:05:20,959 number, greater than 0. 98 00:05:20,959 --> 00:05:24,319 So that would be, this distance right here is epsilon. 99 00:05:24,319 --> 00:05:27,810 This distance there is epsilon. 100 00:05:27,810 --> 00:05:30,480 And for any epsilon you give me, any real number-- so this 101 00:05:30,480 --> 00:05:36,810 is, this would be l plus epsilon right here, this would 102 00:05:36,810 --> 00:05:43,030 be l minus epsilon right here --the epsilon delta definition 103 00:05:43,029 --> 00:05:48,029 of this says that no matter what epsilon one you give me, I 104 00:05:48,029 --> 00:05:51,649 can always specify a distance around a. 105 00:05:51,649 --> 00:05:54,000 And I'll call that delta. 106 00:05:54,000 --> 00:05:57,709 I can always specify a distance around a. 107 00:05:57,709 --> 00:06:02,319 So let's say this is delta less than a, and this 108 00:06:02,319 --> 00:06:04,439 is delta more than a. 109 00:06:04,439 --> 00:06:05,365 This is the letter delta. 110 00:06:05,365 --> 00:06:09,970 111 00:06:09,970 --> 00:06:15,680 Where as long as you pick an x that's within a plus delta and 112 00:06:15,680 --> 00:06:19,439 a minus delta, as long as the x is within here, I can guarantee 113 00:06:19,439 --> 00:06:23,160 you that the f of x, the corresponding f of x is going 114 00:06:23,160 --> 00:06:24,350 to be within your range. 115 00:06:24,350 --> 00:06:26,060 And if you think about it this makes sense right? 116 00:06:26,060 --> 00:06:29,629 It's essentially saying, I can get you as close as you want to 117 00:06:29,629 --> 00:06:32,980 this limit point just by-- and when I say as close as you 118 00:06:32,980 --> 00:06:36,430 want, you define what you want by giving me an epsilon; on 119 00:06:36,430 --> 00:06:38,939 it's a little bit of a game --and I can get you as close as 120 00:06:38,939 --> 00:06:43,000 you want to that limit point by giving you a range around the 121 00:06:43,000 --> 00:06:44,680 point that x is approaching. 122 00:06:44,680 --> 00:06:49,420 And as long as you pick an x value that's within this range 123 00:06:49,420 --> 00:06:52,569 around a, long as you pick an x value around there, I can 124 00:06:52,569 --> 00:06:55,439 guarantee you that f of x will be within the range 125 00:06:55,439 --> 00:06:57,290 you specify. 126 00:06:57,290 --> 00:07:01,270 Just make this a little bit more concrete, let's say you 127 00:07:01,269 --> 00:07:04,490 say, I want f of x to be within 0.5-- let's just you know, make 128 00:07:04,490 --> 00:07:05,379 everything concrete numbers. 129 00:07:05,379 --> 00:07:11,750 Let's say this is the number 2 and let's say this is number 1. 130 00:07:11,750 --> 00:07:16,574 So we're saying that the limit as x approaches 1 of f of x-- I 131 00:07:16,574 --> 00:07:18,879 haven't defined f of x, but it looks like a line with the hole 132 00:07:18,879 --> 00:07:21,480 right there, is equal to 2. 133 00:07:21,480 --> 00:07:23,819 This means that you can give me any number. 134 00:07:23,819 --> 00:07:27,379 Let's say you want to try it out for a couple of examples. 135 00:07:27,379 --> 00:07:30,219 Let's say you say I want f of x to be within point-- let me do 136 00:07:30,220 --> 00:07:35,680 a different color --I want f of x to be within 0.5 of 2. 137 00:07:35,680 --> 00:07:39,970 I want f of x to be between 2.5 and 1.5. 138 00:07:39,970 --> 00:07:45,650 Then I could say, OK, as long as you pick an x within-- I 139 00:07:45,649 --> 00:07:48,189 don't know, it could be arbitrarily close but as long 140 00:07:48,189 --> 00:07:50,920 as you pick an x that's --let's say it works for this function 141 00:07:50,920 --> 00:07:57,790 that's between, I don't know, 0.9 and 1.1. 142 00:07:57,790 --> 00:08:02,980 So in this case the delta from our limit point is only 0.1. 143 00:08:02,980 --> 00:08:09,319 As long as you pick an x that's within 0.1 of this point, or 1, 144 00:08:09,319 --> 00:08:13,639 I can guarantee you that your f of x is going to 145 00:08:13,639 --> 00:08:15,740 lie in that range. 146 00:08:15,740 --> 00:08:17,220 So hopefully you get a little bit of a sense of that. 147 00:08:17,220 --> 00:08:19,750 Let me define that with the actual epsilon delta, and this 148 00:08:19,750 --> 00:08:22,579 is what you'll actually see in your mat textbook, and then 149 00:08:22,579 --> 00:08:24,109 we'll do a couple of examples. 150 00:08:24,110 --> 00:08:26,730 And just to be clear, that was just a specific example. 151 00:08:26,730 --> 00:08:29,870 You gave me one epsilon and I gave you a delta that worked. 152 00:08:29,870 --> 00:08:36,269 But by definition if this is true, or if someone writes 153 00:08:36,269 --> 00:08:40,289 this, they're saying it doesn't just work for one specific 154 00:08:40,289 --> 00:08:42,899 instance, it works for any number you give me. 155 00:08:42,899 --> 00:08:48,799 You can say I want to be within one millionth of, you know, or 156 00:08:48,799 --> 00:08:52,179 ten to the negative hundredth power of 2, you know, super 157 00:08:52,179 --> 00:08:55,589 close to 2, and I can always give you a range around this 158 00:08:55,590 --> 00:09:00,269 point where as long as you pick an x in that range, f of x will 159 00:09:00,269 --> 00:09:03,539 always be within this range that you specify, within that 160 00:09:03,539 --> 00:09:08,240 were you know, one trillionth of a unit away from 161 00:09:08,240 --> 00:09:09,470 the limit point. 162 00:09:09,470 --> 00:09:11,269 And of course, the one thing I can't guarantee is what 163 00:09:11,269 --> 00:09:12,759 happens when x is equal to a. 164 00:09:12,759 --> 00:09:15,580 I'm just saying as long as you pick an x that's within my 165 00:09:15,580 --> 00:09:17,950 range but not on a, it'll work. 166 00:09:17,950 --> 00:09:21,720 Your f of x will show up to be within the range you specify. 167 00:09:21,720 --> 00:09:23,680 And just to make the math clear-- because I've been 168 00:09:23,679 --> 00:09:26,250 speaking only in words so far --and this is what we see the 169 00:09:26,250 --> 00:09:33,460 textbook: it says look, you give me any epsilon 170 00:09:33,460 --> 00:09:35,810 greater than 0. 171 00:09:35,809 --> 00:09:37,389 Anyway, this is a definition, right? 172 00:09:37,389 --> 00:09:41,730 If someone writes this they mean that you can give them any 173 00:09:41,730 --> 00:09:52,800 epsilon greater than 0, and then they'll give you a delta-- 174 00:09:52,799 --> 00:09:56,589 remember your epsilon is how close you want f of x to be 175 00:09:56,590 --> 00:09:57,759 to your limit point, right? 176 00:09:57,759 --> 00:10:00,529 It's a range around f of x --they'll give you a delta 177 00:10:00,529 --> 00:10:04,860 which is a range around a, right? 178 00:10:04,860 --> 00:10:05,519 Let me write this. 179 00:10:05,519 --> 00:10:11,829 So limit as approaches a of f of x is equal to l. 180 00:10:11,830 --> 00:10:15,210 So they'll give you a delta where as long as x is no more 181 00:10:15,210 --> 00:10:23,025 than delta-- So the distance between x and a, so if we pick 182 00:10:23,024 --> 00:10:27,949 an x here-- let me do another color --if we pick an x here, 183 00:10:27,950 --> 00:10:31,340 the distance between that value and a, as long as one, that's 184 00:10:31,340 --> 00:10:34,840 greater than 0 so that x doesn't show up on top of a, 185 00:10:34,840 --> 00:10:37,980 because its function might be undefined at that point. 186 00:10:37,980 --> 00:10:40,750 But as long as the distance between x and a is greater 187 00:10:40,750 --> 00:10:45,399 than 0 and less than this x range that they gave you, 188 00:10:45,399 --> 00:10:46,449 it's less than delta. 189 00:10:46,450 --> 00:10:49,930 So as long as you take an x, you know if I were to zoom the 190 00:10:49,929 --> 00:10:55,679 x-axis right here-- this is a and so this distance right here 191 00:10:55,679 --> 00:10:59,239 would be delta, and this distance right here would be 192 00:10:59,240 --> 00:11:03,919 delta --as long as you pick an x value that falls here-- so as 193 00:11:03,919 --> 00:11:07,519 long as you pick that x value or this x value or this x value 194 00:11:07,519 --> 00:11:10,559 --as long as you pick one of those x values, I can guarantee 195 00:11:10,559 --> 00:11:17,009 you that the distance between your function and the limit 196 00:11:17,009 --> 00:11:19,669 point, so the distance between you know, when you take one of 197 00:11:19,669 --> 00:11:23,459 these x values and you evaluate f of x at that point, that the 198 00:11:23,460 --> 00:11:27,170 distance between that f of x and the limit point is 199 00:11:27,169 --> 00:11:31,559 going to be less than the number you gave them. 200 00:11:31,559 --> 00:11:36,469 And if you think of, it seems very complicated, and I have 201 00:11:36,470 --> 00:11:38,690 mixed feelings about where this is included in most 202 00:11:38,690 --> 00:11:39,640 calculus curriculums. 203 00:11:39,639 --> 00:11:42,345 It's included in like the, you know, the third week before you 204 00:11:42,345 --> 00:11:44,670 even learn derivatives, and it's kind of this very mathy 205 00:11:44,669 --> 00:11:47,559 and rigorous thing to think about, and you know, it tends 206 00:11:47,559 --> 00:11:49,719 to derail a lot of students and a lot of people I don't think 207 00:11:49,720 --> 00:11:53,009 get a lot of the intuition behind it, but it is 208 00:11:53,009 --> 00:11:54,049 mathematically rigorous. 209 00:11:54,049 --> 00:11:56,909 And I think it is very valuable once you study you know, more 210 00:11:56,909 --> 00:11:58,909 advanced calculus or become a math major. 211 00:11:58,909 --> 00:12:01,329 But with that said, this does make a lot of sense 212 00:12:01,330 --> 00:12:02,160 intuitively, right? 213 00:12:02,159 --> 00:12:05,549 Because before we were talking about, look you know, I can get 214 00:12:05,549 --> 00:12:12,944 you as close as x approaches this value f of x is going 215 00:12:12,945 --> 00:12:13,960 to approach this value. 216 00:12:13,960 --> 00:12:17,620 And the way we mathematically define it is, you say Sal, 217 00:12:17,620 --> 00:12:19,970 I want to be super close. 218 00:12:19,970 --> 00:12:22,180 I want the distance to be f of x [UNINTELLIGIBLE]. 219 00:12:22,179 --> 00:12:25,639 And I want it to be 0.000000001, then I can always 220 00:12:25,639 --> 00:12:29,539 give you a distance around x where this will be true. 221 00:12:29,539 --> 00:12:31,319 And I'm all out of time in this video. 222 00:12:31,320 --> 00:12:34,260 In the next video I'll do some examples where I prove the 223 00:12:34,259 --> 00:12:38,120 limits, where I prove some limit statements using 224 00:12:38,120 --> 00:12:39,330 this definition. 225 00:12:39,330 --> 00:12:43,370 And hopefully you know, when we use some tangible numbers, this 226 00:12:43,370 --> 00:12:45,440 definition will make a little bit more sense. 227 00:12:45,440 --> 00:12:47,270 See you in the next video. 228 00:12:47,269 --> 00:12:47,500