Name: Epsilon-delta definition of limits
Uploaded: 2022-07-05T14:15:25.000Z
Duration: 6 min 59 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

Adverbs of degree

with a somewhat rigorous definition of what a limit is,

where we say when you say that the limit of f

of x as x approaches C is equal L, you're really saying--

and this is the somewhat rigorous definition--

want to L by making x sufficiently close to C.

So let's see if we can put a little bit of meat on it.

So instead of saying as close as you want,

let's call that some positive number epsilon.

So I'm just going to use the Greek letter epsilon right over

You tell me how close you want f of x to be to L.

And you do this by giving me a positive number that we call

epsilon, which is really how close you want

f of x to be to L. So you give a positive number epsilon.

And epsilon is how close do you want to be?

that says that you want f of x to be within 0.01 of epsilon.

I'm going to find you another positive number which we'll

call delta-- the lowercase delta, the Greek letter

delta-- such that where if x is within delta of C,

then f of x will be within epsilon of our limit.

So let's see if these are really saying the same thing.

In this yellow definition right over here,

as you want to L by making x sufficiently close to C.

made as a little bit more of a game, is doing the same thing.

Someone is saying how close they want f of x to be to L

and the burden is then to find a delta where as long

It's saying look, if we are constraining x in such a way

f of x to be within epsilon of our limit.

This point right over here is our limit plus epsilon.

And this right over here might be our limit minus epsilon.

I think I can get your f of x within this range of our limit.

And I can do that by defining a range around C.

And I could visually look at this boundary.

But I could even go narrower than that boundary.

So I'll find you some delta so that if you

take any x in the range C minus delta to C

plus delta-- and maybe the function's not even

that maybe aren't C, but are getting very close.

If you find any x in that range, f of those

x's are going to be as close as you want to your limit.

They're going to be within the range L plus epsilon or L

Another way of saying this is you give me an epsilon,

So let me write this in a little bit more math notation.

that's kind of the first part of the game--

So what's another way of saying that x is within delta of C?

This statement is true for any x that's within delta of C.

The difference between the two is going to be less than delta.

So that if you pick an x that is in this range between C

are the x's that satisfy that right over here, then--

the distance between your f of x and your limit--

and this is just the distance between the f of x

and the limit, it's going to be less than epsilon.

So all this is saying is, if the limit truly does exist,

it truly is L, is if you give me any positive number epsilon,

it could be super, super small one, we can find a delta.

is within delta of C, that's all this statement

is saying that the distance between x and C

Subtitles ListPlay Video

Epsilon-delta definition of limits

positive

exact

burden

exist