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