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

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