1 00:00:14,320 --> 00:00:17,312 The last several videos have been about the idea of a derivative, 2 00:00:17,312 --> 00:00:20,940 and before moving on to integrals I want to take some time to talk about limits. 3 00:00:21,660 --> 00:00:24,820 To be honest, the idea of a limit is not really anything new. 4 00:00:25,160 --> 00:00:28,699 If you know what the word approach means you pretty much already know what a limit is. 5 00:00:29,039 --> 00:00:32,301 You could say it's a matter of assigning fancy notation to 6 00:00:32,301 --> 00:00:35,619 the intuitive idea of one value that gets closer to another. 7 00:00:36,439 --> 00:00:39,659 But there are a few reasons to devote a full video to this topic. 8 00:00:40,280 --> 00:00:43,683 For one thing, it's worth showing how the way I've been describing 9 00:00:43,683 --> 00:00:46,732 derivatives so far lines up with the formal definition of a 10 00:00:46,732 --> 00:00:50,239 derivative as it's typically presented in most courses and textbooks. 11 00:00:50,920 --> 00:00:55,103 I want to give you a little confidence that thinking in terms of dx and df 12 00:00:55,103 --> 00:00:59,286 as concrete non-zero nudges is not just some trick for building intuition, 13 00:00:59,286 --> 00:01:03,359 it's backed up by the formal definition of a derivative in all its rigor. 14 00:01:04,260 --> 00:01:08,033 I also want to shed light on what exactly mathematicians mean when 15 00:01:08,033 --> 00:01:11,920 they say approach in terms of the epsilon-delta definition of limits. 16 00:01:12,519 --> 00:01:16,579 Then we'll finish off with a clever trick for computing limits called L'Hopital's rule. 17 00:01:17,799 --> 00:01:21,700 So, first things first, let's take a look at the formal definition of the derivative. 18 00:01:22,319 --> 00:01:25,430 As a reminder, when you have some function f of x, 19 00:01:25,430 --> 00:01:29,762 to think about its derivative at a particular input, maybe x equals 2, 20 00:01:29,762 --> 00:01:33,605 you start by imagining nudging that input some little dx away, 21 00:01:33,605 --> 00:01:36,900 and looking at the resulting change to the output, df. 22 00:01:37,959 --> 00:01:41,466 The ratio df divided by dx, which can be nicely thought of 23 00:01:41,466 --> 00:01:46,757 as the rise over run slope between the starting point on the graph and the nudged point, 24 00:01:46,757 --> 00:01:48,719 is almost what the derivative is. 25 00:01:49,099 --> 00:01:53,959 The actual derivative is whatever this ratio approaches as dx approaches 0. 26 00:01:55,000 --> 00:01:59,123 Just to spell out what's meant there, that nudge to the output 27 00:01:59,123 --> 00:02:05,013 df is the difference between f at the starting input plus dx and f at the starting input, 28 00:02:05,013 --> 00:02:07,500 the change to the output caused by dx. 29 00:02:08,680 --> 00:02:14,341 To express that you want to find what this ratio approaches as dx approaches 0, 30 00:02:14,341 --> 00:02:17,879 you write lim for limit, with dx arrow 0 below it. 31 00:02:18,960 --> 00:02:21,891 You'll almost never see terms with a lowercase 32 00:02:21,890 --> 00:02:24,759 d like dx inside a limit expression like this. 33 00:02:25,319 --> 00:02:28,076 Instead, the standard is to use a different variable, 34 00:02:28,076 --> 00:02:31,039 something like delta x, or commonly h for whatever reason. 35 00:02:31,860 --> 00:02:35,505 The way I like to think of it is that terms with this lowercase 36 00:02:35,504 --> 00:02:40,174 d in the typical derivative expression have built into them this idea of a limit, 37 00:02:40,175 --> 00:02:43,080 the idea that dx is supposed to eventually go to 0. 38 00:02:44,659 --> 00:02:47,609 In a sense, this left hand side here, df over dx, 39 00:02:47,609 --> 00:02:51,207 the ratio we've been thinking about for the past few videos, 40 00:02:51,206 --> 00:02:55,866 is just shorthand for what the right hand side here spells out in more detail, 41 00:02:55,866 --> 00:03:00,939 writing out exactly what we mean by df, and writing out this limit process explicitly. 42 00:03:01,620 --> 00:03:05,265 This right hand side here is the formal definition of a derivative, 43 00:03:05,264 --> 00:03:08,159 as you would commonly see it in any calculus textbook. 44 00:03:08,759 --> 00:03:11,228 And if you'll pardon me for a small rant here, 45 00:03:11,229 --> 00:03:15,800 I want to emphasize that nothing about this right hand side references the paradoxical 46 00:03:15,800 --> 00:03:17,640 idea of an infinitely small change. 47 00:03:18,259 --> 00:03:19,959 The point of limits is to avoid that. 48 00:03:20,620 --> 00:03:23,026 This value h is the exact same thing as the dx 49 00:03:23,026 --> 00:03:25,280 I've been referencing throughout the series. 50 00:03:25,900 --> 00:03:32,280 It's a nudge to the input of f with some non-zero, finitely small size, like 0.001. 51 00:03:33,099 --> 00:03:37,699 It's just that we're analyzing what happens for arbitrarily small choices of h. 52 00:03:38,580 --> 00:03:43,438 In fact, the only reason people introduce a new variable name into this formal 53 00:03:43,437 --> 00:03:48,295 definition, rather than just using dx, is to be extra clear that these changes 54 00:03:48,295 --> 00:03:53,400 to the input are just ordinary numbers that have nothing to do with infinitesimals. 55 00:03:54,379 --> 00:03:59,060 There are others who like to interpret this dx as an infinitely small change, 56 00:03:59,060 --> 00:04:02,719 whatever Or to just say that dx and df are nothing more than 57 00:04:02,719 --> 00:04:05,419 symbols that we shouldn't take too seriously. 58 00:04:06,219 --> 00:04:09,479 But by now in the series, you know I'm not really a fan of either of those views. 59 00:04:10,020 --> 00:04:14,231 I think you can and should interpret dx as a concrete, finitely small nudge, 60 00:04:14,231 --> 00:04:18,500 just so long as you remember to ask what happens when that thing approaches 0. 61 00:04:19,420 --> 00:04:23,071 For one thing, and I hope the past few videos have helped convince you of this, 62 00:04:23,071 --> 00:04:27,180 that helps to build stronger intuition for where the rules of calculus actually come from. 63 00:04:27,180 --> 00:04:29,900 But it's not just some trick for building intuitions. 64 00:04:30,459 --> 00:04:34,129 Everything I've been saying about derivatives with this concrete, 65 00:04:34,129 --> 00:04:38,911 finitely small nudge philosophy is just a translation of this formal definition we're 66 00:04:38,911 --> 00:04:40,079 staring at right now. 67 00:04:41,040 --> 00:04:44,677 Long story short, the big fuss about limits is that they let us 68 00:04:44,677 --> 00:04:48,540 avoid talking about infinitely small changes by instead asking what 69 00:04:48,540 --> 00:04:52,519 happens as the size of some small change to our variable approaches 0. 70 00:04:53,279 --> 00:04:56,129 And this brings us to goal number 2, understanding 71 00:04:56,129 --> 00:04:59,259 exactly what it means for one value to approach another. 72 00:05:00,439 --> 00:05:07,139 For example, consider the function 2 plus h cubed minus 2 cubed all divided by h. 73 00:05:08,480 --> 00:05:12,273 This happens to be the expression that pops out when you unravel 74 00:05:12,273 --> 00:05:16,183 the definition of a derivative of x cubed evaluated at x equals 2, 75 00:05:16,182 --> 00:05:19,860 but let's just think of it as any old function with an input h. 76 00:05:20,439 --> 00:05:23,303 Its graph is this nice continuous looking parabola, 77 00:05:23,303 --> 00:05:27,379 which would make sense because it's a cubic term divided by a linear term. 78 00:05:28,199 --> 00:05:32,204 But actually, if you think about what's going on at h equals 0, 79 00:05:32,204 --> 00:05:36,460 plugging that in you would get 0 divided by 0, which is not defined. 80 00:05:37,420 --> 00:05:40,247 So really, this graph has a hole at that point, 81 00:05:40,247 --> 00:05:45,139 and you have to exaggerate to draw that hole, often with an empty circle like this. 82 00:05:45,139 --> 00:05:47,839 But keep in mind, the function is perfectly well 83 00:05:47,839 --> 00:05:50,319 defined for inputs as close to 0 as you want. 84 00:05:51,259 --> 00:05:55,702 Wouldn't you agree that as h approaches 0, the corresponding output, 85 00:05:55,702 --> 00:05:58,279 the height of this graph, approaches 12? 86 00:05:59,160 --> 00:06:01,580 It doesn't matter which side you come at it from. 87 00:06:03,740 --> 00:06:08,199 That limit of this ratio as h approaches 0 is equal to 12. 88 00:06:09,360 --> 00:06:12,742 But imagine you're a mathematician inventing calculus, 89 00:06:12,742 --> 00:06:17,480 and someone skeptically asks you, well, what exactly do you mean by approach? 90 00:06:18,439 --> 00:06:21,218 That would be kind of an annoying question, I mean, come on, 91 00:06:21,218 --> 00:06:24,180 we all know what it means for one value to get closer to another. 92 00:06:24,939 --> 00:06:28,533 But let's start thinking about ways you might be able to answer that person, 93 00:06:28,533 --> 00:06:29,699 completely unambiguously. 94 00:06:30,939 --> 00:06:34,187 For a given range of inputs within some distance of 0, 95 00:06:34,187 --> 00:06:39,028 excluding the forbidden point 0 itself, look at all of the corresponding outputs, 96 00:06:39,028 --> 00:06:42,040 all possible heights of the graph above that range. 97 00:06:42,860 --> 00:06:47,281 As the range of input values closes in more and more tightly around 0, 98 00:06:47,281 --> 00:06:51,639 that range of output values closes in more and more closely around 12. 99 00:06:52,420 --> 00:06:57,280 And importantly, the size of that range of output values can be made as small as you want. 100 00:06:59,019 --> 00:07:02,526 As a counter example, consider a function that looks like this, 101 00:07:02,526 --> 00:07:06,199 which is also not defined at 0, but kind of jumps up at that point. 102 00:07:06,959 --> 00:07:11,599 When you approach h equals 0 from the right, the function approaches the value 2, 103 00:07:11,600 --> 00:07:14,600 but as you come at it from the left, it approaches 1. 104 00:07:15,540 --> 00:07:20,043 Since there's not a single clear, unambiguous value that this function 105 00:07:20,043 --> 00:07:24,420 approaches as h approaches 0, the limit is not defined at that point. 106 00:07:25,160 --> 00:07:30,066 One way to think of this is that when you look at any range of inputs around 0, 107 00:07:30,065 --> 00:07:35,033 and consider the corresponding range of outputs, as you shrink that input range, 108 00:07:35,033 --> 00:07:38,959 the corresponding outputs don't narrow in on any specific value. 109 00:07:39,779 --> 00:07:43,909 Instead, those outputs straddle a range that never shrinks smaller than 1, 110 00:07:43,910 --> 00:07:47,380 even as you make that input range as tiny as you could imagine. 111 00:07:48,519 --> 00:07:52,298 This perspective of shrinking an input range around the limiting point, 112 00:07:52,298 --> 00:07:56,866 and seeing whether or not you're restricted in how much that shrinks the output range, 113 00:07:56,867 --> 00:08:00,280 leads to something called the epsilon-delta definition of limits. 114 00:08:01,220 --> 00:08:03,319 Now I should tell you, you could argue that this is 115 00:08:03,319 --> 00:08:05,500 needlessly heavy duty for an introduction to calculus. 116 00:08:06,060 --> 00:08:08,353 Like I said, if you know what the word approach means, 117 00:08:08,353 --> 00:08:11,939 you already know what a limit means, there's nothing new on the conceptual level here. 118 00:08:12,319 --> 00:08:16,283 But this is an interesting glimpse into the field of real analysis, 119 00:08:16,283 --> 00:08:21,356 and gives you a taste for how mathematicians make the intuitive ideas of calculus more 120 00:08:21,357 --> 00:08:22,640 airtight and rigorous. 121 00:08:23,699 --> 00:08:25,339 You've already seen the main idea here. 122 00:08:25,660 --> 00:08:29,622 When a limit exists, you can make this output range as small as you want, 123 00:08:29,622 --> 00:08:33,960 but when the limit doesn't exist, that output range cannot get smaller than some 124 00:08:33,960 --> 00:08:38,780 particular value, no matter how much you shrink the input range around the limiting input. 125 00:08:39,678 --> 00:08:42,372 Let's freeze that same idea a little more precisely, 126 00:08:42,373 --> 00:08:45,879 maybe in the context of this example where the limiting value was 12. 127 00:08:46,779 --> 00:08:50,033 Think about any distance away from 12, where for some reason it's 128 00:08:50,033 --> 00:08:53,139 common to use the Greek letter epsilon to denote that distance. 129 00:08:53,820 --> 00:08:58,040 The intent here is that this distance epsilon is as small as you want. 130 00:08:58,820 --> 00:09:04,701 What it means for the limit to exist is that you will always be able to find a 131 00:09:04,701 --> 00:09:10,135 range of inputs around our limiting point, some distance delta around 0, 132 00:09:10,135 --> 00:09:16,016 so that any input within delta of 0 corresponds to an output within a distance 133 00:09:16,017 --> 00:09:17,060 epsilon of 12. 134 00:09:18,419 --> 00:09:21,154 The key point here is that that's true for any epsilon, 135 00:09:21,154 --> 00:09:24,819 no matter how small, you'll always be able to find the corresponding delta. 136 00:09:25,580 --> 00:09:29,999 In contrast, when a limit does not exist, as in this example here, 137 00:09:29,999 --> 00:09:33,495 you can find a sufficiently small epsilon, like 0.4, 138 00:09:33,495 --> 00:09:39,234 so that no matter how small you make your range around 0, no matter how tiny delta is, 139 00:09:39,234 --> 00:09:43,060 the corresponding range of outputs is just always too big. 140 00:09:43,700 --> 00:09:48,640 There is no limiting output where everything is within a distance epsilon of that output. 141 00:09:54,100 --> 00:09:57,159 So far, this is all pretty theory-heavy, don't you think? 142 00:09:57,679 --> 00:10:00,387 Limits being used to formally define the derivative, 143 00:10:00,388 --> 00:10:04,120 and epsilons and deltas being used to rigorously define the limit itself. 144 00:10:04,899 --> 00:10:08,259 So let's finish things off here with a trick for actually computing limits. 145 00:10:09,100 --> 00:10:12,363 For instance, let's say for some reason you were studying 146 00:10:12,363 --> 00:10:15,740 the function sin of pi times x divided by x squared minus 1. 147 00:10:16,220 --> 00:10:19,240 Maybe this was modeling some kind of dampened oscillation. 148 00:10:20,240 --> 00:10:23,460 When you plot a bunch of points to graph this, it looks pretty continuous. 149 00:10:27,279 --> 00:10:29,480 But there's a problematic value at x equals 1. 150 00:10:30,000 --> 00:10:35,341 When you plug that in, sin of pi is 0, and the denominator also comes out to 0, 151 00:10:35,341 --> 00:10:39,014 so the function is actually not defined at that input, 152 00:10:39,014 --> 00:10:41,620 and the graph should have a hole there. 153 00:10:42,200 --> 00:10:45,511 This also happens at x equals negative 1, but let's just 154 00:10:45,510 --> 00:10:48,939 focus our attention on a single one of these holes for now. 155 00:10:50,019 --> 00:10:53,775 The graph certainly does seem to approach a distinct value at that point, 156 00:10:53,775 --> 00:10:54,639 wouldn't you say? 157 00:10:57,279 --> 00:11:03,004 So you might ask, how exactly do you find what output this approaches as x approaches 1, 158 00:11:03,004 --> 00:11:05,000 since you can't just plug in 1? 159 00:11:07,960 --> 00:11:11,624 Well, one way to approximate it would be to plug in 160 00:11:11,624 --> 00:11:15,360 a number that's just really close to 1, like 1.00001. 161 00:11:16,120 --> 00:11:20,080 Doing that, you'd find that this should be a number around negative 1.57. 162 00:11:21,159 --> 00:11:23,600 But is there a way to know precisely what it is? 163 00:11:23,960 --> 00:11:27,596 Some systematic process to take an expression like this one, 164 00:11:27,596 --> 00:11:32,187 that looks like 0 divided by 0 at some input, and ask what is its limit as x 165 00:11:32,187 --> 00:11:33,500 approaches that input? 166 00:11:36,440 --> 00:11:40,157 After limits, so helpfully let us write the definition for derivatives, 167 00:11:40,157 --> 00:11:44,700 derivatives can actually come back here and return the favor to help us evaluate limits. 168 00:11:45,200 --> 00:11:46,020 Let me show you what I mean. 169 00:11:47,019 --> 00:11:50,367 Here's what the graph of sin of pi times x looks like, 170 00:11:50,368 --> 00:11:53,900 and here's what the graph of x squared minus 1 looks like. 171 00:11:53,899 --> 00:11:56,779 That's a lot to have up on the screen, but just 172 00:11:56,779 --> 00:11:59,419 focus on what's happening around x equals 1. 173 00:12:00,179 --> 00:12:06,301 The point here is that sin of pi times x and x squared minus 1 are both 0 at that point, 174 00:12:06,302 --> 00:12:08,159 they both cross the x axis. 175 00:12:09,000 --> 00:12:14,277 In the same spirit as plugging in a specific value near 1, like 1.00001, 176 00:12:14,277 --> 00:12:20,640 let's zoom in on that point and consider what happens just a tiny nudge dx away from it. 177 00:12:21,299 --> 00:12:26,380 The value sin of pi times x is bumped down, and the value of that nudge, 178 00:12:26,380 --> 00:12:32,159 which was caused by the nudge dx to the input, is what we might call d sin of pi x. 179 00:12:33,039 --> 00:12:37,259 And from our knowledge of derivatives, using the chain rule, 180 00:12:37,259 --> 00:12:41,480 that should be around cosine of pi times x times pi times dx. 181 00:12:42,700 --> 00:12:47,700 Since the starting value was x equals 1, we plug in x equals 1 to that expression. 182 00:12:51,259 --> 00:12:56,706 In other words, the amount that this sin of pi times x graph changes is roughly 183 00:12:56,706 --> 00:13:02,360 proportional to dx, with a proportionality constant equal to cosine of pi times pi. 184 00:13:03,360 --> 00:13:06,645 And cosine of pi, if we think back to our trig knowledge, 185 00:13:06,645 --> 00:13:11,179 is exactly negative 1, so we can write this whole thing as negative pi times dx. 186 00:13:12,220 --> 00:13:18,445 Similarly, the value of the x squared minus 1 graph changes by some dx squared minus 1, 187 00:13:18,445 --> 00:13:23,540 and taking the derivative, the size of that nudge should be 2x times dx. 188 00:13:24,480 --> 00:13:29,407 Again, we were starting at x equals 1, so we plug in x equals 1 to that expression, 189 00:13:29,407 --> 00:13:33,280 meaning the size of that output nudge is about 2 times 1 times dx. 190 00:13:34,919 --> 00:13:41,277 What this means is that for values of x which are just a tiny nudge dx away from 1, 191 00:13:41,278 --> 00:13:46,425 the ratio sin of pi x divided by x squared minus 1 is approximately 192 00:13:46,424 --> 00:13:49,679 negative pi times dx divided by 2 times dx. 193 00:13:50,899 --> 00:13:54,740 The dx's cancel out, so what's left is negative pi over 2. 194 00:13:55,720 --> 00:13:58,591 And importantly, those approximations get more and more 195 00:13:58,591 --> 00:14:01,360 accurate for smaller and smaller choices of dx, right? 196 00:14:02,309 --> 00:14:05,618 This ratio, negative pi over 2, actually tells 197 00:14:05,619 --> 00:14:09,000 us the precise limiting value as x approaches 1. 198 00:14:09,539 --> 00:14:13,106 Remember, what that means is that the limiting height on 199 00:14:13,106 --> 00:14:16,799 our original graph is evidently exactly negative pi over 2. 200 00:14:18,220 --> 00:14:21,601 What happened there is a little subtle, so I want to go through it again, 201 00:14:21,601 --> 00:14:23,340 but this time a little more generally. 202 00:14:24,120 --> 00:14:29,388 Instead of these two specific functions, which are both equal to 0 at x equals 1, 203 00:14:29,388 --> 00:14:34,913 think of any two functions, f of x and g of x, which are both 0 at some common value, 204 00:14:34,913 --> 00:14:35,620 x equals a. 205 00:14:36,279 --> 00:14:39,562 The only constraint is that these have to be functions where you're 206 00:14:39,562 --> 00:14:41,928 able to take a derivative of them at x equals a, 207 00:14:41,928 --> 00:14:45,405 which means they each basically look like a line when you zoom in close 208 00:14:45,405 --> 00:14:46,419 enough to that value. 209 00:14:47,799 --> 00:14:52,392 Even though you can't compute f divided by g at this trouble point, 210 00:14:52,393 --> 00:14:56,514 since both of them equal 0, you can ask about this ratio for 211 00:14:56,514 --> 00:15:00,500 values of x really close to a, the limit as x approaches a. 212 00:15:01,220 --> 00:15:06,200 It's helpful to think of those nearby inputs as just a tiny nudge, dx, away from a. 213 00:15:06,759 --> 00:15:12,160 The value of f at that nudged point is approximately its derivative, 214 00:15:12,160 --> 00:15:14,979 df over dx, evaluated at a times dx. 215 00:15:15,980 --> 00:15:22,124 Likewise, the value of g at that nudged point is approximately the derivative of g, 216 00:15:22,124 --> 00:15:23,879 evaluated at a times dx. 217 00:15:25,059 --> 00:15:31,059 Near that trouble point, the ratio between the outputs of f and g is actually about the 218 00:15:31,059 --> 00:15:37,059 same as the derivative of f at a times dx, divided by the derivative of g at a times dx. 219 00:15:37,879 --> 00:15:41,119 Those dx's cancel out, so the ratio of f and g near a 220 00:15:41,119 --> 00:15:44,540 is about the same as the ratio between their derivatives. 221 00:15:45,860 --> 00:15:50,307 Because each of those approximations gets more and more accurate for smaller and 222 00:15:50,307 --> 00:15:54,700 smaller nudges, this ratio of derivatives gives the precise value for the limit. 223 00:15:55,539 --> 00:15:58,500 This is a really handy trick for computing a lot of limits. 224 00:15:58,919 --> 00:16:02,937 Whenever you come across some expression that seems to equal 0 divided by 225 00:16:02,937 --> 00:16:06,900 0 when you plug in some particular input, just try taking the derivative 226 00:16:06,900 --> 00:16:10,919 of the top and bottom expressions and plugging in that same trouble input. 227 00:16:13,980 --> 00:16:16,300 This clever trick is called L'Hopital's Rule. 228 00:16:17,240 --> 00:16:20,182 Interestingly, it was actually discovered by Johann Bernoulli, 229 00:16:20,182 --> 00:16:22,844 but L'Hopital was this wealthy dude who essentially paid 230 00:16:22,844 --> 00:16:25,879 Bernoulli for the rights to some of his mathematical discoveries. 231 00:16:26,740 --> 00:16:30,076 Academia is weird back then, but in a very literal way, 232 00:16:30,076 --> 00:16:32,460 it pays to understand these tiny nudges. 233 00:16:34,960 --> 00:16:38,716 Right now, you might be remembering that the definition of a derivative 234 00:16:38,716 --> 00:16:42,264 for a given function comes down to computing the limit of a certain 235 00:16:42,264 --> 00:16:45,657 fraction that looks like 0 divided by 0, so you might think that 236 00:16:45,657 --> 00:16:49,780 L'Hopital's Rule could give us a handy way to discover new derivative formulas. 237 00:16:50,679 --> 00:16:53,473 But that would actually be cheating, since presumably 238 00:16:53,474 --> 00:16:56,320 you don't know what the derivative of the numerator is. 239 00:16:57,019 --> 00:16:59,593 When it comes to discovering derivative formulas, 240 00:16:59,594 --> 00:17:02,374 something we've been doing a fair amount this series, 241 00:17:02,374 --> 00:17:04,640 there is no systematic plug-and-chug method. 242 00:17:05,118 --> 00:17:05,959 But that's a good thing! 243 00:17:06,400 --> 00:17:09,373 Whenever creativity is needed to solve problems like these, 244 00:17:09,373 --> 00:17:11,901 it's a good sign that you're doing something real, 245 00:17:11,901 --> 00:17:15,420 something that might give you a powerful tool to solve future problems. 246 00:17:18,259 --> 00:17:22,907 And speaking of powerful tools, up next I'm going to be talking about what an integral 247 00:17:22,907 --> 00:17:25,686 is, as well as the fundamental theorem of calculus, 248 00:17:25,686 --> 00:17:30,442 another example of where limits can be used to give a clear meaning to a pretty delicate 249 00:17:30,442 --> 00:17:32,099 idea that flirts with infinity. 250 00:17:33,579 --> 00:17:36,818 As you know, most support for this channel comes through Patreon, 251 00:17:36,818 --> 00:17:40,795 and the primary perk for patrons is early access to future series like this one, 252 00:17:40,795 --> 00:17:43,200 where the next one is going to be on probability. 253 00:17:44,259 --> 00:17:47,754 But for those of you who want a more tangible way to flag that 254 00:17:47,755 --> 00:17:51,640 you're part of the community, there is also a small 3blue1brown store. 255 00:17:52,299 --> 00:17:53,960 Links on the screen and in the description. 256 00:17:54,680 --> 00:18:05,081 I'm still debating whether or not to make a preliminary batch of plushie pie creatures, 257 00:18:05,080 --> 00:18:14,064 it depends on how many viewers seem interested in the store more generally, 258 00:18:14,065 --> 00:18:23,875 but let me know in comments what other kinds of things you'd like to see in there. 259 00:18:23,875 --> 00:18:26,240 Thanks for watching!