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The last several videos have been about the idea of a derivative, and before moving on
to integrals, I want to take some time to talk about limits.
To be honest, the idea of a limit is not really anything new. If you know what the word “approach”
means you pretty much already know what a limit is, you could say the rest is a matter
of assigning fancy notation to the intuitive idea of one value getting closer to another.
But there are actually a few reasons to devote a full video to this topic.
For one thing it’s worth showing is how the way I’ve been describing derivatives
so far lines up with the the formal definition of a derivative as it’s typically presented
in most courses and textbooks. I want to give you some confidence that thinking
of terms like dx and df as concrete non-zero nudges is not just some trick for building
intuition; it’s actually backed up by the formal definition of a derivative in all its
rigor. I also want to shed a little light on what
exactly mathematicians mean by “approach”, in terms of something called the "epsilon
delta" definition of limits. Then we’ll finish off with a clever trick
for computing limits called L’Hopital’s rule.
So first thing’s first, let’s take a look at the formal definition of the derivative.
As a reminder, when you have some function f(x), to think about the derivative at a particular
input, maybe x=2, you start by imagining nudging that input by some tiny dx, and looking at
the resulting change to the output, df. The ratio df/dx, which can nicely be thought
of as the rise-over-run slope between the starting point on the graph and the nudged
point, is almost the derivative. The actual derivative is whatever this ratio approaches
as dx approaches 0. Just to spell out what is meant here, that
nudge to the output “df” is is the difference between f(starting-input + dx) and f(starting-input);
the change to the output caused by the nudge dx.
To express that you want to find what this ratio approaches as dx approaches 0, you write
“l-i-m”, for limit, with “dx arrow 0” below it.
Now, you’ll almost never see terms with a lowercase d, like dx, inside a limit like
this. Instead the standard is to use a different variable, like delta-x, or commonly “h”
for some reason. The way I like to think of it is that terms
with this lowercase d in the typical derivative expression have built into them the idea of
a limit, the idea that dx is supposed to eventually approach 0.
So in a sense this lefthand side “df/dx”, the ratio we’ve been thinking about for
the past few videos, is just shorthand for what the righthand side spells out in more
detail, writing out exactly what we mean by df, and writing out the limiting process explicitly.
And that righthand side is the formal definition of a derivative, as you’d commonly see it
in any calculus textbook
Now, if you’ll pardon me for a small rant here, I want to emphasize that nothing about
this righthand side references the paradoxical idea of an “infinitely small” change.
The point of limits is to avoid that. This value h is the exact same thing as the
“dx” I’ve been referencing throughout the series.
It’s a nudge to the input of f with some nonzero, finitely small size, like 0.001,
it’s just that we’re analyzing what happens for arbitrarily small choices of h.
In fact, the only reason people introduce a new variable name into this formal definition,
rather than just using dx, is to be super-extra clear that these changes to the input are
ordinary numbers that have nothing to do with the infinitesimal.
You see, there are others who like to interpret dx as an “infinitely small change”, whatever
that would mean, or to just say that dx and df are nothing more than symbols that shouldn’t
be taken too seriously. But by now in the series, you know that I’m
not really a fan of either of those views, I think you can and should interpret dx as
a concrete, finitely small nudge, just so long as you remember to ask what happens as
it approaches 0. For one thing, and I hope the past few videos
have helped convince you of this, that helps to build a stronger intuition for where the
rules of calculus actually come from. But it’s not just some trick for building
intuitions. Everything I’ve been saying about derivatives with this concrete-finitely-small-nudge
philosophy is just a translation of the formal definition of derivatives.
Long story short, the big fuss about limits is that they let us avoid talking about infinitely
small changes by instead asking what happens as the size of some change to our variable
approaches 0. And that brings us to goal #2: Understanding
exactly it means for one value to approach another.
For example, consider the function [(2+h)3 - 23]/h.
This happens to be the expression that pops out if you unravel the definition for the
derivative of x3 at x=2, but let’s just think of it as any ol’ function with an
input h. Its graph is this nice continuous looking
parabola. But actually, if you think about what’s going at h=0, plugging that in you’d
get 0/0, which is not defined. Just ask siri. So really, this graph has a hole at that point.
You have to exaggerate to draw that hole, often with a little empty circle like this,
but keep in mind the function is perfectly well-defined for inputs as close to 0 as you
want. And wouldn’t you agree that as h approaches
0, the corresponding output, the height of this graph, approaches 12? And it doesn’t
matter which side you come at it from. That the limit of this ratio as h goes to 0 equals
12. But imagine you’re a mathematician inventing
calculus, and someone skeptically asks “well what exactly do you mean by approach?”
That would be an annoying question. I mean, come on, we all know what it means for one
value to get closer to another. But let me show you a way to answer completely
unambiguously. For a given range of inputs within some distance
of 0, excluding the forbidden point 0, look at the corresponding outputs, all possible
heights of the graph above that range. As that range of input values closes in more
and more tightly around 0, the range of output values closes in more and more closely around
12. The size of that range of outputs can be made as small as you want.
As a counterexample, consider a function that looks like this, which is also not defined
at 0, but kind of jumps at that point. As you approach h = 0 from the right, the
function approaches 2, but as you come at 0 from the left, it approaches 1. Since there’s
not a clear, unambiguous value that this function approaches as h approaches 0, the limit is
simply not defined at that point. When you look at any range of inputs around
0, and the corresponding range of outputs, as you shrink that input range the corresponding
outputs don’t narrow in on any specific value. Instead those outputs straddle a range
that never even shrinks smaller than 1, no matter how small your input range.
This perspective of shrinking an input range around the limiting point, and seeing whether
or not you’re restricted in how much that shrinks the output range, leads to something
called the “epsilon delta” definition of limits.
You could argue this needlessly heavy-duty for an introduction to calculus. Like I said,
if you know what the word “approach” means, you know what a limit means, so there’s
nothing new on the conceptual level here. But this is an interesting glimpse into the
field of real analysis, and it gives you a taste for how mathematicians made the intuitive
ideas of calculus fully airtight and rigorous. You’ve already seen the main idea: when
a limit exists, you can make this output range as small as you want; but when the limit doesn’t
exist, that output range can’t get smaller than some value, no matter how much you shrink
the input range around the limiting input. Phrasing that same idea a little more precisely,
maybe in the context of this example where the limiting value was 12, think of any distance
away from 12, where for some reason it’s common to use the greek letter “epsilon”
to denote that distance. And the intent here is that that distance be something as small
as you want. What it means for the limit to exist is that
you can always find a range of inputs around our limiting input, some distance delta away
from 0, so that any input within a distance delta of 0 corresponds to an output with a
distance epsilon of 12. They key point is that this is true for any
epsilon, no matter how small. In contrast, when a limit doesn’t exist,
as in this example, you can find a sufficiently small epsilon, like 0.4, so that no matter
how small you make your range around 0, no matter how tiny delta is, the corresponding
range of outputs is just always too big. There is no limiting output value that they get
arbitrarily close to.
So far this is all pretty theory heavy; limits being used to formally define the derivative,
then epsilons and deltas being used to rigorously define limits themselves. So let’s finish
things off here with a trick for actually computing limits.
For example, let’s say for some reason you were studying the function sin(pi*x)/(x2-1).
Maybe this models some kind of dampened oscillation. When you plot a bunch of points to graph it,
it looks pretty continuous, but there’s a problematic value, x=1.
When you plug that in, sin(pi) is 0, and the denominator is also 0, so the function is
actually not defined there, and the graph should really have a hole there.
This also happens at -1, but let’s just focus our attention on one of these holes
for now. The graph certainly does seem to approach
some distinct value at that point, wouldn’t you say? So you might ask, how do you figure
out what output this approaches as x approaches 1, since you can’t just plug in 1?
Well, one way to approximate it would be to plug in a number very close to 1, like 1.00001.
Doing that, you’d get a number around -1.57. But is there a way to know exactly what it
is? Some systematic process to take an expression like this one, which looks like 0/0 at some
input, and ask what its limit is as x approaches that input?
Well, after limits so helpfully let us write the definition for a derivative, derivatives
can come back to return the favor and help us evaluate limits. Let me show you what I
mean. Here’s the graph of sin(pi*x), and here’s
the graph of x2-1. That’s kind of a lot on screen, but just focus on what’s happening
at x=1. The point here is that sin(pi*x) and x2-1 are both 0 at that point, so they cross
the x-axis. In the same spirit as plugging in a specific
value near 1, like 1.00001, let’s zoom in on that point and consider what happens a
tiny nudge dx away. The value of sin(pi*x) is bumped down, and
the value of that nudge, which was caused by the nudge dx to the input, is what we might
call d(sin(pi*x)). From our knowledge of derivatives, using the
chain rule, that should be around cos(pi*x)*pi*dx. Since the starting value was x=1, we plug
in x=1 to this expression. In other words, the size of the change to
this sin(pi*x) graph is roughly proportional to dx, with proportionality constant cos(pi)*pi.
Since cos(pi) is exactly -1, we can write that as -pi*dx.
Similarly, the value this x2-1 graph has changed by some d(x2-1). And taking the derivative,
the size of that nudge should be 2*x*dx. Again, since we started at x=1, that means the size
of this change is about 2*1*dx. So for values of x which are some tiny value
dx away from 1, the ratio sin(pi*x)/(x2-1) is approximately (-pi*dx) / (2*dx). The dx’s
cancel, so that value is -pi/2. Since these approximations get more and more
accurate for smaller and smaller choices of dx, this ratio -pi/2 actually tells us the
precise limiting value as x approaches 1. Remember, what that means is that the limiting
height on our original graph is evidently exactly -pi/2.
What happened there is a little subtle, so let me show it again, but this time a little
more generally. Instead of these two specific functions, which both equal 0 at x=1, think
of any two functions f(x) and g(x), which are both 0 at some common value x = a.
And these have to be functions where you’re able to take a derivative of them at x = a,
meaning they each basically look like a line when you zoom in close enough to that value.
Even though you can’t compute f divided by g at the trouble point, since both equal
zero, you can ask abou this ratio for values of x very close to a, the limit as x approach
a. And it’s helpful to think of those nearby inputs as a tiny nudge dx away from a.
The value of f at that nudged point is approximately its derivative, df/dx evaluated at a, times
dx. Likewise the the value of g at that nudged point is approximately the derivative of g,
evaluated at a, times dx. So near this 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.
These dx’s cancel, so the ratio of f and g near a is about the same as the ratio between
their derivatives. Since those approximations get more accurate
for smaller nudges, this ratio of derivatives gives the precise value for the limit.
This is a really handy trick for computing a lot of limits. If you come across an expression
that seems to equal 0/0 when you plug in some input, just take the derivative of the top
and bottom expressions, and plug in that trouble input.
This clever trick is called “L'Hôpital's rule”. Interestingly, it was actually discovered
by Johann Bernoulli, but L’Hopital was a wealthy dude who essentially paid Bernoulli
for the rights to some of his mathematical discoveries.
In a very literal way, it pays to understand these tiny nudges.
You might remember that the definition of a derivative for any given function comes
down to computing the limit of a fraction that looks like 0/0, so you might think L’Hopital’s
rule gives a handy way to discover new derivative formulas.
But that would be cheating, since presumably you don’t yet know what the derivative on
the numerator here is. When it comes to discovering derivative formulas,
something we’ve been doing a fair amount this series, there is no systematic plug-and-chug
method. But that’s a good thing. When creativity is required to solve problems like these,
it’s a good sign you’re doing something real; something that might give you a powerful
tool to solve future problems.
Up next, I’ll talk about what an integral is, as well as the fundamental theorem of
calculus, which is another example of where limits are used to help give a clear meaning
to a fairly delicate idea that flirts with infinity.
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