WEBVTT 00:00:14.320 --> 00:00:17.312 The last several videos have been about the idea of a derivative, 00:00:17.312 --> 00:00:20.940 and before moving on to integrals I want to take some time to talk about limits. 00:00:21.660 --> 00:00:24.820 To be honest, the idea of a limit is not really anything new. 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. 00:00:29.039 --> 00:00:32.301 You could say it's a matter of assigning fancy notation to 00:00:32.301 --> 00:00:35.619 the intuitive idea of one value that gets closer to another. 00:00:36.439 --> 00:00:39.659 But there are a few reasons to devote a full video to this topic. 00:00:40.280 --> 00:00:43.683 For one thing, it's worth showing how the way I've been describing 00:00:43.683 --> 00:00:46.732 derivatives so far lines up with the formal definition of a 00:00:46.732 --> 00:00:50.239 derivative as it's typically presented in most courses and textbooks. 00:00:50.920 --> 00:00:55.103 I want to give you a little confidence that thinking in terms of dx and df 00:00:55.103 --> 00:00:59.286 as concrete non-zero nudges is not just some trick for building intuition, 00:00:59.286 --> 00:01:03.359 it's backed up by the formal definition of a derivative in all its rigor. 00:01:04.260 --> 00:01:08.033 I also want to shed light on what exactly mathematicians mean when 00:01:08.033 --> 00:01:11.920 they say approach in terms of the epsilon-delta definition of limits. 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. 00:01:17.799 --> 00:01:21.700 So, first things first, let's take a look at the formal definition of the derivative. 00:01:22.319 --> 00:01:25.430 As a reminder, when you have some function f of x, 00:01:25.430 --> 00:01:29.762 to think about its derivative at a particular input, maybe x equals 2, 00:01:29.762 --> 00:01:33.605 you start by imagining nudging that input some little dx away, 00:01:33.605 --> 00:01:36.900 and looking at the resulting change to the output, df. 00:01:37.959 --> 00:01:41.466 The ratio df divided by dx, which can be nicely thought of 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, 00:01:46.757 --> 00:01:48.719 is almost what the derivative is. 00:01:49.099 --> 00:01:53.959 The actual derivative is whatever this ratio approaches as dx approaches 0. 00:01:55.000 --> 00:01:59.123 Just to spell out what's meant there, that nudge to the output 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, 00:02:05.013 --> 00:02:07.500 the change to the output caused by dx. 00:02:08.680 --> 00:02:14.341 To express that you want to find what this ratio approaches as dx approaches 0, 00:02:14.341 --> 00:02:17.879 you write lim for limit, with dx arrow 0 below it. 00:02:18.960 --> 00:02:21.891 You'll almost never see terms with a lowercase 00:02:21.890 --> 00:02:24.759 d like dx inside a limit expression like this. 00:02:25.319 --> 00:02:28.076 Instead, the standard is to use a different variable, 00:02:28.076 --> 00:02:31.039 something like delta x, or commonly h for whatever reason. 00:02:31.860 --> 00:02:35.505 The way I like to think of it is that terms with this lowercase 00:02:35.504 --> 00:02:40.174 d in the typical derivative expression have built into them this idea of a limit, 00:02:40.175 --> 00:02:43.080 the idea that dx is supposed to eventually go to 0. 00:02:44.659 --> 00:02:47.609 In a sense, this left hand side here, df over dx, 00:02:47.609 --> 00:02:51.207 the ratio we've been thinking about for the past few videos, 00:02:51.206 --> 00:02:55.866 is just shorthand for what the right hand side here spells out in more detail, 00:02:55.866 --> 00:03:00.939 writing out exactly what we mean by df, and writing out this limit process explicitly. 00:03:01.620 --> 00:03:05.265 This right hand side here is the formal definition of a derivative, 00:03:05.264 --> 00:03:08.159 as you would commonly see it in any calculus textbook. 00:03:08.759 --> 00:03:11.228 And if you'll pardon me for a small rant here, 00:03:11.229 --> 00:03:15.800 I want to emphasize that nothing about this right hand side references the paradoxical 00:03:15.800 --> 00:03:17.640 idea of an infinitely small change. 00:03:18.259 --> 00:03:19.959 The point of limits is to avoid that. 00:03:20.620 --> 00:03:23.026 This value h is the exact same thing as the dx 00:03:23.026 --> 00:03:25.280 I've been referencing throughout the series. 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. 00:03:33.099 --> 00:03:37.699 It's just that we're analyzing what happens for arbitrarily small choices of h. 00:03:38.580 --> 00:03:43.438 In fact, the only reason people introduce a new variable name into this formal 00:03:43.437 --> 00:03:48.295 definition, rather than just using dx, is to be extra clear that these changes 00:03:48.295 --> 00:03:53.400 to the input are just ordinary numbers that have nothing to do with infinitesimals. 00:03:54.379 --> 00:03:59.060 There are others who like to interpret this dx as an infinitely small change, 00:03:59.060 --> 00:04:02.719 whatever Or to just say that dx and df are nothing more than 00:04:02.719 --> 00:04:05.419 symbols that we shouldn't take too seriously. 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. 00:04:10.020 --> 00:04:14.231 I think you can and should interpret dx as a concrete, finitely small nudge, 00:04:14.231 --> 00:04:18.500 just so long as you remember to ask what happens when that thing approaches 0. 00:04:19.420 --> 00:04:23.071 For one thing, and I hope the past few videos have helped convince you of this, 00:04:23.071 --> 00:04:27.180 that helps to build stronger intuition for where the rules of calculus actually come from. 00:04:27.180 --> 00:04:29.900 But it's not just some trick for building intuitions. 00:04:30.459 --> 00:04:34.129 Everything I've been saying about derivatives with this concrete, 00:04:34.129 --> 00:04:38.911 finitely small nudge philosophy is just a translation of this formal definition we're 00:04:38.911 --> 00:04:40.079 staring at right now. 00:04:41.040 --> 00:04:44.677 Long story short, the big fuss about limits is that they let us 00:04:44.677 --> 00:04:48.540 avoid talking about infinitely small changes by instead asking what 00:04:48.540 --> 00:04:52.519 happens as the size of some small change to our variable approaches 0. 00:04:53.279 --> 00:04:56.129 And this brings us to goal number 2, understanding 00:04:56.129 --> 00:04:59.259 exactly what it means for one value to approach another. 00:05:00.439 --> 00:05:07.139 For example, consider the function 2 plus h cubed minus 2 cubed all divided by h. 00:05:08.480 --> 00:05:12.273 This happens to be the expression that pops out when you unravel 00:05:12.273 --> 00:05:16.183 the definition of a derivative of x cubed evaluated at x equals 2, 00:05:16.182 --> 00:05:19.860 but let's just think of it as any old function with an input h. 00:05:20.439 --> 00:05:23.303 Its graph is this nice continuous looking parabola, 00:05:23.303 --> 00:05:27.379 which would make sense because it's a cubic term divided by a linear term. 00:05:28.199 --> 00:05:32.204 But actually, if you think about what's going on at h equals 0, 00:05:32.204 --> 00:05:36.460 plugging that in you would get 0 divided by 0, which is not defined. 00:05:37.420 --> 00:05:40.247 So really, this graph has a hole at that point, 00:05:40.247 --> 00:05:45.139 and you have to exaggerate to draw that hole, often with an empty circle like this. 00:05:45.139 --> 00:05:47.839 But keep in mind, the function is perfectly well 00:05:47.839 --> 00:05:50.319 defined for inputs as close to 0 as you want. 00:05:51.259 --> 00:05:55.702 Wouldn't you agree that as h approaches 0, the corresponding output, 00:05:55.702 --> 00:05:58.279 the height of this graph, approaches 12? 00:05:59.160 --> 00:06:01.580 It doesn't matter which side you come at it from. 00:06:03.740 --> 00:06:08.199 That limit of this ratio as h approaches 0 is equal to 12. 00:06:09.360 --> 00:06:12.742 But imagine you're a mathematician inventing calculus, 00:06:12.742 --> 00:06:17.480 and someone skeptically asks you, well, what exactly do you mean by approach? 00:06:18.439 --> 00:06:21.218 That would be kind of an annoying question, I mean, come on, 00:06:21.218 --> 00:06:24.180 we all know what it means for one value to get closer to another. 00:06:24.939 --> 00:06:28.533 But let's start thinking about ways you might be able to answer that person, 00:06:28.533 --> 00:06:29.699 completely unambiguously. 00:06:30.939 --> 00:06:34.187 For a given range of inputs within some distance of 0, 00:06:34.187 --> 00:06:39.028 excluding the forbidden point 0 itself, look at all of the corresponding outputs, 00:06:39.028 --> 00:06:42.040 all possible heights of the graph above that range. 00:06:42.860 --> 00:06:47.281 As the range of input values closes in more and more tightly around 0, 00:06:47.281 --> 00:06:51.639 that range of output values closes in more and more closely around 12. 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. 00:06:59.019 --> 00:07:02.526 As a counter example, consider a function that looks like this, 00:07:02.526 --> 00:07:06.199 which is also not defined at 0, but kind of jumps up at that point. 00:07:06.959 --> 00:07:11.599 When you approach h equals 0 from the right, the function approaches the value 2, 00:07:11.600 --> 00:07:14.600 but as you come at it from the left, it approaches 1. 00:07:15.540 --> 00:07:20.043 Since there's not a single clear, unambiguous value that this function 00:07:20.043 --> 00:07:24.420 approaches as h approaches 0, the limit is not defined at that point. 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, 00:07:30.065 --> 00:07:35.033 and consider the corresponding range of outputs, as you shrink that input range, 00:07:35.033 --> 00:07:38.959 the corresponding outputs don't narrow in on any specific value. 00:07:39.779 --> 00:07:43.909 Instead, those outputs straddle a range that never shrinks smaller than 1, 00:07:43.910 --> 00:07:47.380 even as you make that input range as tiny as you could imagine. 00:07:48.519 --> 00:07:52.298 This perspective of shrinking an input range around the limiting point, 00:07:52.298 --> 00:07:56.866 and seeing whether or not you're restricted in how much that shrinks the output range, 00:07:56.867 --> 00:08:00.280 leads to something called the epsilon-delta definition of limits. 00:08:01.220 --> 00:08:03.319 Now I should tell you, you could argue that this is 00:08:03.319 --> 00:08:05.500 needlessly heavy duty for an introduction to calculus. 00:08:06.060 --> 00:08:08.353 Like I said, if you know what the word approach means, 00:08:08.353 --> 00:08:11.939 you already know what a limit means, there's nothing new on the conceptual level here. 00:08:12.319 --> 00:08:16.283 But this is an interesting glimpse into the field of real analysis, 00:08:16.283 --> 00:08:21.356 and gives you a taste for how mathematicians make the intuitive ideas of calculus more 00:08:21.357 --> 00:08:22.640 airtight and rigorous. 00:08:23.699 --> 00:08:25.339 You've already seen the main idea here. 00:08:25.660 --> 00:08:29.622 When a limit exists, you can make this output range as small as you want, 00:08:29.622 --> 00:08:33.960 but when the limit doesn't exist, that output range cannot get smaller than some 00:08:33.960 --> 00:08:38.780 particular value, no matter how much you shrink the input range around the limiting input. 00:08:39.678 --> 00:08:42.372 Let's freeze that same idea a little more precisely, 00:08:42.373 --> 00:08:45.879 maybe in the context of this example where the limiting value was 12. 00:08:46.779 --> 00:08:50.033 Think about any distance away from 12, where for some reason it's 00:08:50.033 --> 00:08:53.139 common to use the Greek letter epsilon to denote that distance. 00:08:53.820 --> 00:08:58.040 The intent here is that this distance epsilon is as small as you want. 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 00:09:04.701 --> 00:09:10.135 range of inputs around our limiting point, some distance delta around 0, 00:09:10.135 --> 00:09:16.016 so that any input within delta of 0 corresponds to an output within a distance 00:09:16.017 --> 00:09:17.060 epsilon of 12. 00:09:18.419 --> 00:09:21.154 The key point here is that that's true for any epsilon, 00:09:21.154 --> 00:09:24.819 no matter how small, you'll always be able to find the corresponding delta. 00:09:25.580 --> 00:09:29.999 In contrast, when a limit does not exist, as in this example here, 00:09:29.999 --> 00:09:33.495 you can find a sufficiently small epsilon, like 0.4, 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, 00:09:39.234 --> 00:09:43.060 the corresponding range of outputs is just always too big. 00:09:43.700 --> 00:09:48.640 There is no limiting output where everything is within a distance epsilon of that output. 00:09:54.100 --> 00:09:57.159 So far, this is all pretty theory-heavy, don't you think? 00:09:57.679 --> 00:10:00.387 Limits being used to formally define the derivative, 00:10:00.388 --> 00:10:04.120 and epsilons and deltas being used to rigorously define the limit itself. 00:10:04.899 --> 00:10:08.259 So let's finish things off here with a trick for actually computing limits. 00:10:09.100 --> 00:10:12.363 For instance, let's say for some reason you were studying 00:10:12.363 --> 00:10:15.740 the function sin of pi times x divided by x squared minus 1. 00:10:16.220 --> 00:10:19.240 Maybe this was modeling some kind of dampened oscillation. 00:10:20.240 --> 00:10:23.460 When you plot a bunch of points to graph this, it looks pretty continuous. 00:10:27.279 --> 00:10:29.480 But there's a problematic value at x equals 1. 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, 00:10:35.341 --> 00:10:39.014 so the function is actually not defined at that input, 00:10:39.014 --> 00:10:41.620 and the graph should have a hole there. 00:10:42.200 --> 00:10:45.511 This also happens at x equals negative 1, but let's just 00:10:45.510 --> 00:10:48.939 focus our attention on a single one of these holes for now. 00:10:50.019 --> 00:10:53.775 The graph certainly does seem to approach a distinct value at that point, 00:10:53.775 --> 00:10:54.639 wouldn't you say? 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, 00:11:03.004 --> 00:11:05.000 since you can't just plug in 1? 00:11:07.960 --> 00:11:11.624 Well, one way to approximate it would be to plug in 00:11:11.624 --> 00:11:15.360 a number that's just really close to 1, like 1.00001. 00:11:16.120 --> 00:11:20.080 Doing that, you'd find that this should be a number around negative 1.57. 00:11:21.159 --> 00:11:23.600 But is there a way to know precisely what it is? 00:11:23.960 --> 00:11:27.596 Some systematic process to take an expression like this one, 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 00:11:32.187 --> 00:11:33.500 approaches that input? 00:11:36.440 --> 00:11:40.157 After limits, so helpfully let us write the definition for derivatives, 00:11:40.157 --> 00:11:44.700 derivatives can actually come back here and return the favor to help us evaluate limits. 00:11:45.200 --> 00:11:46.020 Let me show you what I mean. 00:11:47.019 --> 00:11:50.367 Here's what the graph of sin of pi times x looks like, 00:11:50.368 --> 00:11:53.900 and here's what the graph of x squared minus 1 looks like. 00:11:53.899 --> 00:11:56.779 That's a lot to have up on the screen, but just 00:11:56.779 --> 00:11:59.419 focus on what's happening around x equals 1. 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, 00:12:06.302 --> 00:12:08.159 they both cross the x axis. 00:12:09.000 --> 00:12:14.277 In the same spirit as plugging in a specific value near 1, like 1.00001, 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. 00:12:21.299 --> 00:12:26.380 The value sin of pi times x is bumped down, and the value of that nudge, 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. 00:12:33.039 --> 00:12:37.259 And from our knowledge of derivatives, using the chain rule, 00:12:37.259 --> 00:12:41.480 that should be around cosine of pi times x times pi times dx. 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. 00:12:51.259 --> 00:12:56.706 In other words, the amount that this sin of pi times x graph changes is roughly 00:12:56.706 --> 00:13:02.360 proportional to dx, with a proportionality constant equal to cosine of pi times pi. 00:13:03.360 --> 00:13:06.645 And cosine of pi, if we think back to our trig knowledge, 00:13:06.645 --> 00:13:11.179 is exactly negative 1, so we can write this whole thing as negative pi times dx. 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, 00:13:18.445 --> 00:13:23.540 and taking the derivative, the size of that nudge should be 2x times dx. 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, 00:13:29.407 --> 00:13:33.280 meaning the size of that output nudge is about 2 times 1 times dx. 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, 00:13:41.278 --> 00:13:46.425 the ratio sin of pi x divided by x squared minus 1 is approximately 00:13:46.424 --> 00:13:49.679 negative pi times dx divided by 2 times dx. 00:13:50.899 --> 00:13:54.740 The dx's cancel out, so what's left is negative pi over 2. 00:13:55.720 --> 00:13:58.591 And importantly, those approximations get more and more 00:13:58.591 --> 00:14:01.360 accurate for smaller and smaller choices of dx, right? 00:14:02.309 --> 00:14:05.618 This ratio, negative pi over 2, actually tells 00:14:05.619 --> 00:14:09.000 us the precise limiting value as x approaches 1. 00:14:09.539 --> 00:14:13.106 Remember, what that means is that the limiting height on 00:14:13.106 --> 00:14:16.799 our original graph is evidently exactly negative pi over 2. 00:14:18.220 --> 00:14:21.601 What happened there is a little subtle, so I want to go through it again, 00:14:21.601 --> 00:14:23.340 but this time a little more generally. 00:14:24.120 --> 00:14:29.388 Instead of these two specific functions, which are both equal to 0 at x equals 1, 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, 00:14:34.913 --> 00:14:35.620 x equals a. 00:14:36.279 --> 00:14:39.562 The only constraint is that these have to be functions where you're 00:14:39.562 --> 00:14:41.928 able to take a derivative of them at x equals a, 00:14:41.928 --> 00:14:45.405 which means they each basically look like a line when you zoom in close 00:14:45.405 --> 00:14:46.419 enough to that value. 00:14:47.799 --> 00:14:52.392 Even though you can't compute f divided by g at this trouble point, 00:14:52.393 --> 00:14:56.514 since both of them equal 0, you can ask about this ratio for 00:14:56.514 --> 00:15:00.500 values of x really close to a, the limit as x approaches a. 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. 00:15:06.759 --> 00:15:12.160 The value of f at that nudged point is approximately its derivative, 00:15:12.160 --> 00:15:14.979 df over dx, evaluated at a times dx. 00:15:15.980 --> 00:15:22.124 Likewise, the value of g at that nudged point is approximately the derivative of g, 00:15:22.124 --> 00:15:23.879 evaluated at a times dx. 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 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. 00:15:37.879 --> 00:15:41.119 Those dx's cancel out, so the ratio of f and g near a 00:15:41.119 --> 00:15:44.540 is about the same as the ratio between their derivatives. 00:15:45.860 --> 00:15:50.307 Because each of those approximations gets more and more accurate for smaller and 00:15:50.307 --> 00:15:54.700 smaller nudges, this ratio of derivatives gives the precise value for the limit. 00:15:55.539 --> 00:15:58.500 This is a really handy trick for computing a lot of limits. 00:15:58.919 --> 00:16:02.937 Whenever you come across some expression that seems to equal 0 divided by 00:16:02.937 --> 00:16:06.900 0 when you plug in some particular input, just try taking the derivative 00:16:06.900 --> 00:16:10.919 of the top and bottom expressions and plugging in that same trouble input. 00:16:13.980 --> 00:16:16.300 This clever trick is called L'Hopital's Rule. 00:16:17.240 --> 00:16:20.182 Interestingly, it was actually discovered by Johann Bernoulli, 00:16:20.182 --> 00:16:22.844 but L'Hopital was this wealthy dude who essentially paid 00:16:22.844 --> 00:16:25.879 Bernoulli for the rights to some of his mathematical discoveries. 00:16:26.740 --> 00:16:30.076 Academia is weird back then, but in a very literal way, 00:16:30.076 --> 00:16:32.460 it pays to understand these tiny nudges. 00:16:34.960 --> 00:16:38.716 Right now, you might be remembering that the definition of a derivative 00:16:38.716 --> 00:16:42.264 for a given function comes down to computing the limit of a certain 00:16:42.264 --> 00:16:45.657 fraction that looks like 0 divided by 0, so you might think that 00:16:45.657 --> 00:16:49.780 L'Hopital's Rule could give us a handy way to discover new derivative formulas. 00:16:50.679 --> 00:16:53.473 But that would actually be cheating, since presumably 00:16:53.474 --> 00:16:56.320 you don't know what the derivative of the numerator is. 00:16:57.019 --> 00:16:59.593 When it comes to discovering derivative formulas, 00:16:59.594 --> 00:17:02.374 something we've been doing a fair amount this series, 00:17:02.374 --> 00:17:04.640 there is no systematic plug-and-chug method. 00:17:05.118 --> 00:17:05.959 But that's a good thing! 00:17:06.400 --> 00:17:09.373 Whenever creativity is needed to solve problems like these, 00:17:09.373 --> 00:17:11.901 it's a good sign that you're doing something real, 00:17:11.901 --> 00:17:15.420 something that might give you a powerful tool to solve future problems. 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 00:17:22.907 --> 00:17:25.686 is, as well as the fundamental theorem of calculus, 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 00:17:30.442 --> 00:17:32.099 idea that flirts with infinity. 00:17:33.579 --> 00:17:36.818 As you know, most support for this channel comes through Patreon, 00:17:36.818 --> 00:17:40.795 and the primary perk for patrons is early access to future series like this one, 00:17:40.795 --> 00:17:43.200 where the next one is going to be on probability. 00:17:44.259 --> 00:17:47.754 But for those of you who want a more tangible way to flag that 00:17:47.755 --> 00:17:51.640 you're part of the community, there is also a small 3blue1brown store. 00:17:52.299 --> 00:17:53.960 Links on the screen and in the description. 00:17:54.680 --> 00:18:05.081 I'm still debating whether or not to make a preliminary batch of plushie pie creatures, 00:18:05.080 --> 00:18:14.064 it depends on how many viewers seem interested in the store more generally, 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. 00:18:23.875 --> 00:18:26.240 Thanks for watching!