Limits, L'Hôpital's rule, and epsilon delta definitions | Chapter 7, Essence of calculus

Summary

This video script delves into the concept of limits in calculus, explaining how they underpin the definition of derivatives. It clarifies the formal definition of a derivative using limits to avoid paradoxical infinitesimals, introduces the epsilon-delta definition for a rigorous understanding of limits, and concludes with L'Hopital's Rule—a trick for computing limits involving indeterminate forms like 0/0. The script aims to build intuition and confidence in calculus concepts, emphasizing the importance of limits in calculus.

Takeaways

• 📘 The concept of a limit is intuitively similar to the idea of something 'approaching' another value.
• 🔄 The formal definition of a derivative aligns with the intuitive understanding of limits and the concrete use of 'dx'.
• 📐 The epsilon-delta definition of limits provides a rigorous mathematical framework to describe 'approaching' values.
• 🚫 Limits avoid the paradoxical concept of infinitesimals by focusing on the behavior as changes approach zero.
• 📉 The limit of a function at a certain point is the value the function output approaches, even if the function is undefined at that point.
• 🔍 The epsilon-delta definition helps to determine if a limit exists by examining the behavior of input and output ranges around the limit point.
• 📌 The concept of limits is fundamental to calculus and is used to define derivatives formally.
• 🔄 L'Hopital's Rule is a clever trick for computing limits when faced with expressions that result in '0/0' indeterminate forms.
• 📚 L'Hopital's Rule involves taking the derivatives of the numerator and denominator and evaluating them at the problematic point.
• 🎓 Understanding limits and L'Hopital's Rule is crucial for a deeper comprehension of calculus and its applications.

Q & A

What is the main focus of the video script?

The main focus of the video script is to explain the concept of limits in calculus, its formal definition, and its relation to the derivative. It also aims to clarify the epsilon-delta definition of limits and introduces L'Hopital's rule for computing limits.

How does the script define the derivative?

The script defines the derivative as the limit of the ratio of the change in the output (df) to the change in the input (dx) as dx approaches 0.

What is the difference between dx and delta x in the context of limits?

In the context of limits, dx is often used informally to represent a small change in the input, while delta x is a more formal term used in limit expressions to represent that same small change. The use of delta x emphasizes that these changes are ordinary numbers and not infinitesimals.

Why is the epsilon-delta definition of limits important?

The epsilon-delta definition of limits is important because it provides a rigorous and unambiguous way to define what it means for one value to approach another, which is essential for a formal understanding of calculus.

What is L'Hopital's rule and how does it relate to limits?

L'Hopital's rule is a method for evaluating limits of indeterminate forms, such as 0/0 or ∞/∞, by taking the derivatives of the numerator and denominator and evaluating the limit of the resulting expression. It relates to limits by providing a way to find the limit of a ratio when direct substitution leads to an indeterminate form.

How does the script explain the concept of approaching a limit?

The script explains approaching a limit by discussing how the output values of a function get closer to a certain value as the input values get closer to a specific point, excluding that point itself.

Why does the script emphasize that the derivative does not involve infinitely small changes?

The script emphasizes that the derivative does not involve infinitely small changes to avoid paradoxical ideas and to align with the rigorous formal definition of a derivative, which is based on limits and does not require the concept of infinitesimals.

What is the purpose of using different variable names like h or delta x instead of dx in formal limit definitions?

Using different variable names like h or delta x in formal limit definitions serves to clarify that these are ordinary numbers with non-zero, finitely small sizes, and to distinguish them from the informal use of dx which is often associated with infinitesimals.

How does the script use the concept of limits to explain the behavior of functions at undefined points?

The script uses the concept of limits to explain that even at points where a function is undefined (like 0/0), we can still discuss the behavior of the function by looking at how the function's output values approach a certain value as the input values get closer to the undefined point.

What is the significance of the function 2 + h^3 - 2^3 / h in the script?

The function 2 + h^3 - 2^3 / h is significant because it demonstrates the concept of limits by showing how the output of the function approaches 12 as h approaches 0, even though the function is undefined at h = 0 due to division by zero.

How does the script connect the idea of limits with the concept of derivatives?

The script connects the idea of limits with the concept of derivatives by showing that the definition of a derivative involves finding the limit of a ratio as the change in the input approaches zero, thus linking the intuitive understanding of derivatives with their formal definition.

Outlines

00:00

📘 Introduction to Limits and Derivatives

The paragraph introduces the concept of limits in calculus, emphasizing that it's not a new idea but rather a formalization of the intuitive notion of one value approaching another. It connects this idea to the derivative, explaining that the derivative can be thought of as the ratio of the change in the output (df) to the change in the input (dx) as dx approaches zero. The paragraph also clarifies that the formal definition of a derivative avoids the concept of infinitesimals by focusing on the limit process. It sets the stage for explaining the epsilon-delta definition of limits and introduces L'Hopital's rule for computing limits.

05:00

📐 Epsilon-Delta Definition of Limits

This section delves into the epsilon-delta definition of limits, which provides a rigorous way to describe what it means for a function to approach a certain value. It contrasts a function with a well-defined limit at a point with one that does not have a limit, illustrating how the output values do not converge to a single value when the input values approach the point from both sides. The explanation uses the concept of shrinking ranges of input and output values to show how the limit can be defined mathematically, offering a deeper understanding of the formalism behind limits in calculus.

10:05

🔍 L'Hopital's Rule for Computing Limits

The paragraph explains L'Hopital's Rule, a technique for evaluating limits of indeterminate forms such as 0/0 or ∞/∞. It uses the example of sin(pi*x)/(x^2 - 1) to demonstrate how taking the derivatives of the numerator and denominator and evaluating them at the point of interest can yield the limit. The explanation shows the process of approximating the function's behavior near the point of interest using derivatives, leading to a precise value for the limit. The paragraph also touches on the historical context of L'Hopital's Rule and its significance in calculus.

15:07

🌟 Derivatives, L'Hopital's Rule, and Future Topics

In the final paragraph, the focus is on the practical application of derivatives and L'Hopital's Rule. It mentions that while there is no simple formula for discovering new derivatives, the process requires creativity and understanding, which are valuable in problem-solving. The paragraph also hints at upcoming topics, such as integrals and the fundamental theorem of calculus, which will further explore the use of limits. Additionally, it mentions the support for the educational channel through Patreon and the 3blue1brown store, inviting viewers to engage with the community and provide feedback on potential merchandise.

Keywords

💡Derivative

A derivative in calculus represents the rate at which a function's output changes as its input changes. It is a fundamental concept that helps in understanding how quantities are interconnected. In the video, the derivative is discussed in the context of limits and is defined formally as the limit of the ratio of the infinitesimal change in the output (df) to the infinitesimal change in the input (dx) as dx approaches zero.

💡Limit

The concept of a limit is central to calculus and describes the value that a function or sequence 'approaches' as the input or index approaches some value. In the video, limits are used to define derivatives and are explained through the epsilon-delta definition, which provides a precise way to talk about one value approaching another.

💡Epsilon-Delta Definition

The epsilon-delta definition is a formal way to define what it means for a function to approach a certain value. It uses two variables, epsilon and delta, to express the idea that for any small change in the input (within delta), the output changes by a correspondingly small amount (within epsilon). This concept is crucial for rigorously defining limits in calculus.

💡L'Hopital's Rule

L'Hopital's Rule is a technique for evaluating limits of indeterminate forms, such as 0/0 or ∞/∞, by taking the derivatives of the numerator and denominator. The video explains this rule as a clever trick that uses the concept of derivatives to compute limits, particularly when direct substitution leads to an indeterminate form.

💡Indeterminate Form

An indeterminate form arises in calculus when direct substitution of a limit results in an expression that does not conclusively determine the limit's value, such as 0/0 or ∞/∞. The video discusses how L'Hopital's Rule can be applied to evaluate limits when faced with such forms.

💡Function

A function in mathematics is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. The video uses functions to illustrate the concepts of derivatives and limits, showing how the output of a function changes in relation to changes in its input.

💡Nudge

In the context of the video, a 'nudge' refers to a small change made to the input of a function to observe the resulting change in the output. This concept is used to explain how derivatives are calculated by considering the ratio of the change in output (df) to the change in input (dx) as dx approaches zero.

💡Formal Definition

A formal definition in mathematics is a precise and unambiguous statement of how a concept is to be understood. The video emphasizes the formal definition of a derivative, which is expressed in terms of limits, to contrast with more intuitive explanations that use the idea of 'nudges'.

💡Infinitesimals

Infinitesimals are quantities that are closer to zero than any number with a finite number of decimal places, yet are not zero. The video discusses the debate over whether to interpret dx as an infinitesimal or as a finite, non-zero quantity, ultimately advocating for the latter to build intuition for calculus.

💡Trigonometric Functions

Trigonometric functions, like sine and cosine, relate the angles of a triangle to the ratios of its sides. The video uses the sine function as part of an example to illustrate how L'Hopital's Rule can be applied to find a limit involving a trigonometric expression.

💡Chain Rule

The chain rule is a fundamental rule of calculus for differentiating composite functions. It is used in the video to explain how the derivative of a composite function, such as sin(pi*x), can be found by multiplying the derivative of the outer function by the derivative of the inner function.

Highlights

The concept of a limit is intuitively understood as one value approaching another.

Derivatives can be formally defined using limits to avoid paradoxical ideas of infinitesimals.

The derivative is the limit of the ratio df/dx as dx approaches 0.

The formal definition of a derivative involves the limit notation without terms like 'dx'.

The lowercase 'd' in derivatives implies a limit process where the change approaches zero.

The epsilon-delta definition of limits provides a rigorous understanding of 'approach'.

Limits help to define what it means for a function to approach a certain value.

L'Hopital's rule is a clever trick for computing limits involving indeterminate forms like 0/0.

L'Hopital's rule involves taking the derivatives of the numerator and denominator and evaluating the limit.

The epsilon-delta definition can be visualized by shrinking input ranges and observing output ranges.

A limit exists if for every epsilon, there exists a delta such that the output range is within epsilon of the limit.

When a limit does not exist, the output range cannot be shrunk to a single value regardless of how small the input range is.

The concept of limits is used to define derivatives without resorting to infinitesimals.

L'Hopital's rule can be applied when a function's graph has a hole, indicating an undefined value at a certain point.

The rule provides a systematic way to find the limit of a ratio when direct substitution results in 0/0.

The epsilon-delta definition is a rigorous mathematical concept introduced to formalize the idea of limits.

The use of limits in calculus helps to avoid discussing infinitely small quantities.

L'Hopital's rule is named after Guillaume de l'Hôpital, who financed the work of Johann Bernoulli.

There is no systematic method for discovering new derivative formulas, requiring creativity.

The next topic to be discussed is integrals and the fundamental theorem of calculus, which also utilizes limits.

Transcripts

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 it's a matter of assigning fancy notation to

the intuitive idea of one value that gets closer to another.

But there are a few reasons to devote a full video to this topic.

For one thing, it's worth showing how the way 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.

I want to give you a little confidence that thinking in terms of dx and df

as concrete non-zero nudges is not just some trick for building intuition,

it's backed up by the formal definition of a derivative in all its rigor.

I also want to shed light on what exactly mathematicians mean when

they say approach in terms of 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 things first, let's take a look at the formal definition of the derivative.

As a reminder, when you have some function f of x,

to think about its derivative at a particular input, maybe x equals 2,

you start by imagining nudging that input some little dx away,

and looking at the resulting change to the output, df.

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.

The actual derivative is whatever this ratio approaches as dx approaches 0.

Just to spell out what's meant there, 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.

To express that you want to find what this ratio approaches as dx approaches 0,

you write lim for limit, with dx arrow 0 below it.

You'll almost never see terms with a lowercase

d like dx inside a limit expression like this.

Instead, the standard is to use a different variable,

something like delta x, or commonly h for whatever 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 this idea of a limit,

the idea that dx is supposed to eventually go to 0.

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,

writing out exactly what we mean by df, and writing out this limit process explicitly.

This right hand side here is the formal definition of a derivative,

as you would commonly see it in any calculus textbook.

And if you'll pardon me for a small rant here,

I want to emphasize that nothing about this right hand 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 non-zero, 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 extra clear that these changes

to the input are just ordinary numbers that have nothing to do with infinitesimals.

There are others who like to interpret this dx as an infinitely small change,

whatever Or to just say that dx and df are nothing more than

symbols that we shouldn't take too seriously.

But by now in the series, you know 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 when that thing approaches 0.

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.

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 this formal definition we're

staring at right now.

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 small change to our variable approaches 0.

And this brings us to goal number 2, understanding

exactly what it means for one value to approach another.

For example, consider the function 2 plus h cubed minus 2 cubed all divided by h.

This happens to be the expression that pops out when you unravel

the definition of a derivative of x cubed evaluated at x equals 2,

but let's just think of it as any old function with an input h.

Its graph is this nice continuous looking parabola,

which would make sense because it's a cubic term divided by a linear term.

But actually, if you think about what's going on at h equals 0,

plugging that in you would get 0 divided by 0, which is not defined.

So really, this graph has a hole at that point,

and you have to exaggerate to draw that hole, often with an empty circle like this.

But keep in mind, the function is perfectly well

defined for inputs as close to 0 as you want.

Wouldn't you agree that as h approaches 0, the corresponding output,

the height of this graph, approaches 12?

It doesn't matter which side you come at it from.

That limit of this ratio as h approaches 0 is equal to 12.

But imagine you're a mathematician inventing calculus,

and someone skeptically asks you, well, what exactly do you mean by approach?

That would be kind of an annoying question, I mean, come on,

we all know what it means for one value to get closer to another.

But let's start thinking about ways you might be able to answer that person,

completely unambiguously.

For a given range of inputs within some distance of 0,

excluding the forbidden point 0 itself, look at all of the corresponding outputs,

all possible heights of the graph above that range.

As the range of input values closes in more and more tightly around 0,

that range of output values closes in more and more closely around 12.

And importantly, the size of that range of output values can be made as small as you want.

As a counter example, consider a function that looks like this,

which is also not defined at 0, but kind of jumps up at that point.

When you approach h equals 0 from the right, the function approaches the value 2,

but as you come at it from the left, it approaches 1.

Since there's not a single clear, unambiguous value that this function

approaches as h approaches 0, the limit is not defined at that point.

One way to think of this is that when you look at any range of inputs around 0,

and consider 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 shrinks smaller than 1,

even as you make that input range as tiny as you could imagine.

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.

Now I should tell you, you could argue that this is

needlessly heavy duty for an introduction to calculus.

Like I said, if you know what the word approach means,

you already know what a limit means, there's nothing new on the conceptual level here.

But this is an interesting glimpse into the field of real analysis,

and gives you a taste for how mathematicians make the intuitive ideas of calculus more

airtight and rigorous.

You've already seen the main idea here.

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 cannot get smaller than some

particular value, no matter how much you shrink the input range around the limiting input.

Let's freeze that same idea a little more precisely,

maybe in the context of this example where the limiting value was 12.

Think about any distance away from 12, where for some reason it's

common to use the Greek letter epsilon to denote that distance.

The intent here is that this distance epsilon is as small as you want.

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 delta around 0,

so that any input within delta of 0 corresponds to an output within a distance

epsilon of 12.

The key point here is that that's true for any epsilon,

no matter how small, you'll always be able to find the corresponding delta.

In contrast, when a limit does not exist, as in this example here,

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 where everything is within a distance epsilon of that output.

So far, this is all pretty theory-heavy, don't you think?

Limits being used to formally define the derivative,

and epsilons and deltas being used to rigorously define the limit itself.

So let's finish things off here with a trick for actually computing limits.

For instance, let's say for some reason you were studying

the function sin of pi times x divided by x squared minus 1.

Maybe this was modeling some kind of dampened oscillation.

When you plot a bunch of points to graph this, it looks pretty continuous.

But there's a problematic value at x equals 1.

When you plug that in, sin of pi is 0, and the denominator also comes out to 0,

so the function is actually not defined at that input,

and the graph should have a hole there.

This also happens at x equals negative 1, but let's just

focus our attention on a single one of these holes for now.

The graph certainly does seem to approach a distinct value at that point,

wouldn't you say?

So you might ask, how exactly do you find 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 that's just really close to 1, like 1.00001.

Doing that, you'd find that this should be a number around negative 1.57.

But is there a way to know precisely what it is?

Some systematic process to take an expression like this one,

that looks like 0 divided by 0 at some input, and ask what is its limit as x

approaches that input?

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.

Let me show you what I mean.

Here's what the graph of sin of pi times x looks like,

and here's what the graph of x squared minus 1 looks like.

That's a lot to have up on the screen, but just

focus on what's happening around x equals 1.

The point here is that sin of pi times x and x squared minus 1 are both 0 at that point,

they both 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 just a tiny nudge dx away from it.

The value sin of pi times 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 of pi x.

And from our knowledge of derivatives, using the chain rule,

that should be around cosine of pi times x times pi times dx.

Since the starting value was x equals 1, we plug in x equals 1 to that expression.

In other words, the amount that this sin of pi times x graph changes is roughly

proportional to dx, with a proportionality constant equal to cosine of pi times pi.

And cosine of pi, if we think back to our trig knowledge,

is exactly negative 1, so we can write this whole thing as negative pi times dx.

Similarly, the value of the x squared minus 1 graph changes by some dx squared minus 1,

and taking the derivative, the size of that nudge should be 2x times dx.

Again, we were starting at x equals 1, so we plug in x equals 1 to that expression,

meaning the size of that output nudge is about 2 times 1 times dx.

What this means is that for values of x which are just a tiny nudge dx away from 1,

the ratio sin of pi x divided by x squared minus 1 is approximately

negative pi times dx divided by 2 times dx.

The dx's cancel out, so what's left is negative pi over 2.

And importantly, those approximations get more and more

accurate for smaller and smaller choices of dx, right?

This ratio, negative pi over 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 negative pi over 2.

What happened there is a little subtle, so I want to go through it again,

but this time a little more generally.

Instead of these two specific functions, which are both equal to 0 at x equals 1,

think of any two functions, f of x and g of x, which are both 0 at some common value,

x equals a.

The only constraint is that these have to be functions where you're

able to take a derivative of them at x equals a,

which means 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 this trouble point,

since both of them equal 0, you can ask about this ratio for

values of x really close to a, the limit as x approaches a.

It's helpful to think of those nearby inputs as just a tiny nudge, dx, away from a.

The value of f at that nudged point is approximately its derivative,

df over dx, evaluated at a times dx.

Likewise, the value of g at that nudged point is approximately the derivative of g,

evaluated at a times dx.

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.

Those dx's cancel out, so the ratio of f and g near a

is about the same as the ratio between their derivatives.

Because each of those approximations gets more and more accurate for smaller and

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.

Whenever you come across some expression that seems to equal 0 divided by

0 when you plug in some particular input, just try taking the derivative

of the top and bottom expressions and plugging in that same trouble input.

This clever trick is called L'Hopital's Rule.

Interestingly, it was actually discovered by Johann Bernoulli,

but L'Hopital was this wealthy dude who essentially paid

Bernoulli for the rights to some of his mathematical discoveries.

Academia is weird back then, but in a very literal way,

it pays to understand these tiny nudges.

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 0 divided by 0, so you might think that

L'Hopital's Rule could give us a handy way to discover new derivative formulas.

But that would actually be cheating, since presumably

you don't know what the derivative of the numerator 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!

Whenever creativity is needed to solve problems like these,

it's a good sign that you're doing something real,

something that might give you a powerful tool to solve future problems.

And speaking of powerful tools, up next I'm going to be talking about what an integral

is, as well as the fundamental theorem of calculus,

another example of where limits can be used to give a clear meaning to a pretty delicate

idea that flirts with infinity.

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