[00:14] The last several videos have been about the idea of a derivative,
[00:17] and before moving on to integrals I want to take some time to talk about limits.
[00:21] To be honest, the idea of a limit is not really anything new.
[00:25] If you know what the word approach means you pretty much already know what a limit is.
[00:29] You could say it's a matter of assigning fancy notation to
[00:32] the intuitive idea of one value that gets closer to another.
[00:36] But there are a few reasons to devote a full video to this topic.
[00:40] For one thing, it's worth showing how the way I've been describing
[00:43] derivatives so far lines up with the formal definition of a
[00:46] derivative as it's typically presented in most courses and textbooks.
[00:50] I want to give you a little confidence that thinking in terms of dx and df
[00:55] as concrete non-zero nudges is not just some trick for building intuition,
[00:59] it's backed up by the formal definition of a derivative in all its rigor.
[01:04] I also want to shed light on what exactly mathematicians mean when
[01:08] they say approach in terms of the epsilon-delta definition of limits.
[01:12] Then we'll finish off with a clever trick for computing limits called L'Hopital's rule.
[01:17] So, first things first, let's take a look at the formal definition of the derivative.
[01:22] As a reminder, when you have some function f of x,
[01:25] to think about its derivative at a particular input, maybe x equals 2,
[01:29] you start by imagining nudging that input some little dx away,
[01:33] and looking at the resulting change to the output, df.
[01:37] The ratio df divided by dx, which can be nicely thought of
[01:41] as the rise over run slope between the starting point on the graph and the nudged point,
[01:46] is almost what the derivative is.
[01:49] The actual derivative is whatever this ratio approaches as dx approaches 0.
[01:55] Just to spell out what's meant there, that nudge to the output
[01:59] df is the difference between f at the starting input plus dx and f at the starting input,
[02:05] the change to the output caused by dx.
[02:08] To express that you want to find what this ratio approaches as dx approaches 0,
[02:14] you write lim for limit, with dx arrow 0 below it.
[02:18] You'll almost never see terms with a lowercase
[02:21] d like dx inside a limit expression like this.
[02:25] Instead, the standard is to use a different variable,
[02:28] something like delta x, or commonly h for whatever reason.
[02:31] The way I like to think of it is that terms with this lowercase
[02:35] d in the typical derivative expression have built into them this idea of a limit,
[02:40] the idea that dx is supposed to eventually go to 0.
[02:44] In a sense, this left hand side here, df over dx,
[02:47] the ratio we've been thinking about for the past few videos,
[02:51] is just shorthand for what the right hand side here spells out in more detail,
[02:55] writing out exactly what we mean by df, and writing out this limit process explicitly.
[03:01] This right hand side here is the formal definition of a derivative,
[03:05] as you would commonly see it in any calculus textbook.
[03:08] And if you'll pardon me for a small rant here,
[03:11] I want to emphasize that nothing about this right hand side references the paradoxical
[03:15] idea of an infinitely small change.
[03:18] The point of limits is to avoid that.
[03:20] This value h is the exact same thing as the dx
[03:23] I've been referencing throughout the series.
[03:25] It's a nudge to the input of f with some non-zero, finitely small size, like 0.001.
[03:33] It's just that we're analyzing what happens for arbitrarily small choices of h.
[03:38] In fact, the only reason people introduce a new variable name into this formal
[03:43] definition, rather than just using dx, is to be extra clear that these changes
[03:48] to the input are just ordinary numbers that have nothing to do with infinitesimals.
[03:54] There are others who like to interpret this dx as an infinitely small change,
[03:59] whatever Or to just say that dx and df are nothing more than
[04:02] symbols that we shouldn't take too seriously.
[04:06] But by now in the series, you know I'm not really a fan of either of those views.
[04:10] I think you can and should interpret dx as a concrete, finitely small nudge,
[04:14] just so long as you remember to ask what happens when that thing approaches 0.
[04:19] For one thing, and I hope the past few videos have helped convince you of this,
[04:23] that helps to build stronger intuition for where the rules of calculus actually come from.
[04:27] But it's not just some trick for building intuitions.
[04:30] Everything I've been saying about derivatives with this concrete,
[04:34] finitely small nudge philosophy is just a translation of this formal definition we're
[04:38] staring at right now.
[04:41] Long story short, the big fuss about limits is that they let us
[04:44] avoid talking about infinitely small changes by instead asking what
[04:48] happens as the size of some small change to our variable approaches 0.
[04:53] And this brings us to goal number 2, understanding
[04:56] exactly what it means for one value to approach another.
[05:00] For example, consider the function 2 plus h cubed minus 2 cubed all divided by h.
[05:08] This happens to be the expression that pops out when you unravel
[05:12] the definition of a derivative of x cubed evaluated at x equals 2,
[05:16] but let's just think of it as any old function with an input h.
[05:20] Its graph is this nice continuous looking parabola,
[05:23] which would make sense because it's a cubic term divided by a linear term.
[05:28] But actually, if you think about what's going on at h equals 0,
[05:32] plugging that in you would get 0 divided by 0, which is not defined.
[05:37] So really, this graph has a hole at that point,
[05:40] and you have to exaggerate to draw that hole, often with an empty circle like this.
[05:45] But keep in mind, the function is perfectly well
[05:47] defined for inputs as close to 0 as you want.
[05:51] Wouldn't you agree that as h approaches 0, the corresponding output,
[05:55] the height of this graph, approaches 12?
[05:59] It doesn't matter which side you come at it from.
[06:03] That limit of this ratio as h approaches 0 is equal to 12.
[06:09] But imagine you're a mathematician inventing calculus,
[06:12] and someone skeptically asks you, well, what exactly do you mean by approach?
[06:18] That would be kind of an annoying question, I mean, come on,
[06:21] we all know what it means for one value to get closer to another.
[06:24] But let's start thinking about ways you might be able to answer that person,
[06:28] completely unambiguously.
[06:30] For a given range of inputs within some distance of 0,
[06:34] excluding the forbidden point 0 itself, look at all of the corresponding outputs,
[06:39] all possible heights of the graph above that range.
[06:42] As the range of input values closes in more and more tightly around 0,
[06:47] that range of output values closes in more and more closely around 12.
[06:52] And importantly, the size of that range of output values can be made as small as you want.
[06:59] As a counter example, consider a function that looks like this,
[07:02] which is also not defined at 0, but kind of jumps up at that point.
[07:06] When you approach h equals 0 from the right, the function approaches the value 2,
[07:11] but as you come at it from the left, it approaches 1.
[07:15] Since there's not a single clear, unambiguous value that this function
[07:20] approaches as h approaches 0, the limit is not defined at that point.
[07:25] One way to think of this is that when you look at any range of inputs around 0,
[07:30] and consider the corresponding range of outputs, as you shrink that input range,
[07:35] the corresponding outputs don't narrow in on any specific value.
[07:39] Instead, those outputs straddle a range that never shrinks smaller than 1,
[07:43] even as you make that input range as tiny as you could imagine.
[07:48] This perspective of shrinking an input range around the limiting point,
[07:52] and seeing whether or not you're restricted in how much that shrinks the output range,
[07:56] leads to something called the epsilon-delta definition of limits.
[08:01] Now I should tell you, you could argue that this is
[08:03] needlessly heavy duty for an introduction to calculus.
[08:06] Like I said, if you know what the word approach means,
[08:08] you already know what a limit means, there's nothing new on the conceptual level here.
[08:12] But this is an interesting glimpse into the field of real analysis,
[08:16] and gives you a taste for how mathematicians make the intuitive ideas of calculus more
[08:21] airtight and rigorous.
[08:23] You've already seen the main idea here.
[08:25] When a limit exists, you can make this output range as small as you want,
[08:29] but when the limit doesn't exist, that output range cannot get smaller than some
[08:33] particular value, no matter how much you shrink the input range around the limiting input.
[08:39] Let's freeze that same idea a little more precisely,
[08:42] maybe in the context of this example where the limiting value was 12.
[08:46] Think about any distance away from 12, where for some reason it's
[08:50] common to use the Greek letter epsilon to denote that distance.
[08:53] The intent here is that this distance epsilon is as small as you want.
[08:58] What it means for the limit to exist is that you will always be able to find a
[09:04] range of inputs around our limiting point, some distance delta around 0,
[09:10] so that any input within delta of 0 corresponds to an output within a distance
[09:16] epsilon of 12.
[09:18] The key point here is that that's true for any epsilon,
[09:21] no matter how small, you'll always be able to find the corresponding delta.
[09:25] In contrast, when a limit does not exist, as in this example here,
[09:29] you can find a sufficiently small epsilon, like 0.4,
[09:33] so that no matter how small you make your range around 0, no matter how tiny delta is,
[09:39] the corresponding range of outputs is just always too big.
[09:43] There is no limiting output where everything is within a distance epsilon of that output.
[09:54] So far, this is all pretty theory-heavy, don't you think?
[09:57] Limits being used to formally define the derivative,
[10:00] and epsilons and deltas being used to rigorously define the limit itself.
[10:04] So let's finish things off here with a trick for actually computing limits.
[10:09] For instance, let's say for some reason you were studying
[10:12] the function sin of pi times x divided by x squared minus 1.
[10:16] Maybe this was modeling some kind of dampened oscillation.
[10:20] When you plot a bunch of points to graph this, it looks pretty continuous.
[10:27] But there's a problematic value at x equals 1.
[10:30] When you plug that in, sin of pi is 0, and the denominator also comes out to 0,
[10:35] so the function is actually not defined at that input,
[10:39] and the graph should have a hole there.
[10:42] This also happens at x equals negative 1, but let's just
[10:45] focus our attention on a single one of these holes for now.
[10:50] The graph certainly does seem to approach a distinct value at that point,
[10:53] wouldn't you say?
[10:57] So you might ask, how exactly do you find what output this approaches as x approaches 1,
[11:03] since you can't just plug in 1?
[11:07] Well, one way to approximate it would be to plug in
[11:11] a number that's just really close to 1, like 1.00001.
[11:16] Doing that, you'd find that this should be a number around negative 1.57.
[11:21] But is there a way to know precisely what it is?
[11:23] Some systematic process to take an expression like this one,
[11:27] that looks like 0 divided by 0 at some input, and ask what is its limit as x
[11:32] approaches that input?
[11:36] After limits, so helpfully let us write the definition for derivatives,
[11:40] derivatives can actually come back here and return the favor to help us evaluate limits.
[11:45] Let me show you what I mean.
[11:47] Here's what the graph of sin of pi times x looks like,
[11:50] and here's what the graph of x squared minus 1 looks like.
[11:53] That's a lot to have up on the screen, but just
[11:56] focus on what's happening around x equals 1.
[12:00] The point here is that sin of pi times x and x squared minus 1 are both 0 at that point,
[12:06] they both cross the x axis.
[12:09] In the same spirit as plugging in a specific value near 1, like 1.00001,
[12:14] let's zoom in on that point and consider what happens just a tiny nudge dx away from it.
[12:21] The value sin of pi times x is bumped down, and the value of that nudge,
[12:26] which was caused by the nudge dx to the input, is what we might call d sin of pi x.
[12:33] And from our knowledge of derivatives, using the chain rule,
[12:37] that should be around cosine of pi times x times pi times dx.
[12:42] Since the starting value was x equals 1, we plug in x equals 1 to that expression.
[12:51] In other words, the amount that this sin of pi times x graph changes is roughly
[12:56] proportional to dx, with a proportionality constant equal to cosine of pi times pi.
[13:03] And cosine of pi, if we think back to our trig knowledge,
[13:06] is exactly negative 1, so we can write this whole thing as negative pi times dx.
[13:12] Similarly, the value of the x squared minus 1 graph changes by some dx squared minus 1,
[13:18] and taking the derivative, the size of that nudge should be 2x times dx.
[13:24] Again, we were starting at x equals 1, so we plug in x equals 1 to that expression,
[13:29] meaning the size of that output nudge is about 2 times 1 times dx.
[13:34] What this means is that for values of x which are just a tiny nudge dx away from 1,
[13:41] the ratio sin of pi x divided by x squared minus 1 is approximately
[13:46] negative pi times dx divided by 2 times dx.
[13:50] The dx's cancel out, so what's left is negative pi over 2.
[13:55] And importantly, those approximations get more and more
[13:58] accurate for smaller and smaller choices of dx, right?
[14:02] This ratio, negative pi over 2, actually tells
[14:05] us the precise limiting value as x approaches 1.
[14:09] Remember, what that means is that the limiting height on
[14:13] our original graph is evidently exactly negative pi over 2.
[14:18] What happened there is a little subtle, so I want to go through it again,
[14:21] but this time a little more generally.
[14:24] Instead of these two specific functions, which are both equal to 0 at x equals 1,
[14:29] think of any two functions, f of x and g of x, which are both 0 at some common value,
[14:34] x equals a.
[14:36] The only constraint is that these have to be functions where you're
[14:39] able to take a derivative of them at x equals a,
[14:41] which means they each basically look like a line when you zoom in close
[14:45] enough to that value.
[14:47] Even though you can't compute f divided by g at this trouble point,
[14:52] since both of them equal 0, you can ask about this ratio for
[14:56] values of x really close to a, the limit as x approaches a.
[15:01] It's helpful to think of those nearby inputs as just a tiny nudge, dx, away from a.
[15:06] The value of f at that nudged point is approximately its derivative,
[15:12] df over dx, evaluated at a times dx.
[15:15] Likewise, the value of g at that nudged point is approximately the derivative of g,
[15:22] evaluated at a times dx.
[15:25] Near that trouble point, the ratio between the outputs of f and g is actually about the
[15:31] same as the derivative of f at a times dx, divided by the derivative of g at a times dx.
[15:37] Those dx's cancel out, so the ratio of f and g near a
[15:41] is about the same as the ratio between their derivatives.
[15:45] Because each of those approximations gets more and more accurate for smaller and
[15:50] smaller nudges, this ratio of derivatives gives the precise value for the limit.
[15:55] This is a really handy trick for computing a lot of limits.
[15:58] Whenever you come across some expression that seems to equal 0 divided by
[16:02] 0 when you plug in some particular input, just try taking the derivative
[16:06] of the top and bottom expressions and plugging in that same trouble input.
[16:13] This clever trick is called L'Hopital's Rule.
[16:17] Interestingly, it was actually discovered by Johann Bernoulli,
[16:20] but L'Hopital was this wealthy dude who essentially paid
[16:22] Bernoulli for the rights to some of his mathematical discoveries.
[16:26] Academia is weird back then, but in a very literal way,
[16:30] it pays to understand these tiny nudges.
[16:34] Right now, you might be remembering that the definition of a derivative
[16:38] for a given function comes down to computing the limit of a certain
[16:42] fraction that looks like 0 divided by 0, so you might think that
[16:45] L'Hopital's Rule could give us a handy way to discover new derivative formulas.
[16:50] But that would actually be cheating, since presumably
[16:53] you don't know what the derivative of the numerator is.
[16:57] When it comes to discovering derivative formulas,
[16:59] something we've been doing a fair amount this series,
[17:02] there is no systematic plug-and-chug method.
[17:05] But that's a good thing!
[17:06] Whenever creativity is needed to solve problems like these,
[17:09] it's a good sign that you're doing something real,
[17:11] something that might give you a powerful tool to solve future problems.
[17:18] And speaking of powerful tools, up next I'm going to be talking about what an integral
[17:22] is, as well as the fundamental theorem of calculus,
[17:25] another example of where limits can be used to give a clear meaning to a pretty delicate
[17:30] idea that flirts with infinity.
[17:33] As you know, most support for this channel comes through Patreon,
[17:36] and the primary perk for patrons is early access to future series like this one,
[17:40] where the next one is going to be on probability.
[17:44] But for those of you who want a more tangible way to flag that
[17:47] you're part of the community, there is also a small 3blue1brown store.
[17:52] Links on the screen and in the description.
[17:54] I'm still debating whether or not to make a preliminary batch of plushie pie creatures,
[18:05] it depends on how many viewers seem interested in the store more generally,
[18:14] but let me know in comments what other kinds of things you'd like to see in there.
[18:23] Thanks for watching!