Name: Formal definition of limits Part 2: building the idea | AP Calculus AB | Khan Academy
Uploaded: 2022-07-05T14:15:18.000Z
Duration: 6 min 6 s
Description: VoiceTubeの動画で発音を聞きながら英語表現を覚えよう！学べる英語：

命令形

Let's try to come up with a mathematically rigorous

definition for what this statement means.

of x as x approaches c is equal to L. So let's

say that this means that you can get f of x as close

So another way of saying this is, if you tell me, hey,

I want to get my f of x to be within 0.5 of this limit.

Then you're telling me if this limit is actually true,

you should be able to hand me a value around c.

That if x is within that range, then f of x

is definitely going to be as close to L as I desire.

So let me draw that out to make it a little bit clearer.

So let's say that this right over here is my y-axis.

I'm going to draw a slightly different function, just so we

can really focus on what's going on around here.

The range is around c, and the range is around L. So that's x.

So let's say our function looks, is doing something like,

let's say it does something like, let's see, I

You can always find a limit even where is defined.

But let's say our function looks something like that.

And it can have a little kink in it, the way I drew it.

So this is the point where there's a hole.

So it even has a little kink in it, just like that.

And what we want to do is prove that the limit,

as x, the limit of f of x-- and let me make it clear,

of x-- we want to get an idea for what this definition is

If we're claiming that the limit of f of x, as x approaches c,

So conceptually, we get the gist already.

We already get the gist that this right over here is L.

Well, it's saying that you can get f of x as close

your f of x is going to be in the range that you want.

So let me actually go through that exercise.

So someone comes up to you and says, well, OK.

claiming the limit of f of x as x approaches c is equal to L.

I want to get f of x within 0.5 of L. So this right over here

that if you take any x within that range, your f of x

is always going to fall in this range that you care about.

And so you look at this-- and obviously we haven't explicitly

But you can even eyeball it, the way this function is defined.

And you say that this value, just the way

And let's say that this value right over here is c plus 0.25.

as you get x within 0.25 of c, so as long

as your x's are sitting someplace over here,

the corresponding f of x is going to sit in the range

Maybe instead of saying within the 0.5, I want to get within

And then you'd have to do this exercise again and find

you would have to be able to do this for any range

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Formal definition of limits Part 2: building the idea | AP Calculus AB | Khan Academy

slightly

sense

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