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• - [Tutor] In this video, we're going to try to understand

• limits of composite functions, or at least a way

• of thinking about limits of composite functions

• and in particular, we're gonna think about the case

• where we're trying to find the limit as x approaches a,

• of f of g of x

• and we're going to see under certain circumstances,

• this is going to be equal to f of the limit,

• the limit as x approaches a of g of x

• and what are those circumstances you are asking?

• Well, this is going to be true

• if and only if two things are true,

• first of all, this limit needs to exist.

• So the limit as x approaches a of g of x needs to exist,

• so that needs to exist and then on top of that,

• the function f needs to be continuous at this point

• and f continuous at L.

• So let's look at some examples

• and see if we can apply this idea

• or see if we can't apply it.

• So here I have two functions,

• that are graphically represented right over here,

• let me make sure I have enough space for them

• and what we see on the left-hand side is our function f

• and what we see on the right-hand side is our function g.

• So first let's figure out what is the limit

• as x approaches negative three

• of f of g of x.

• Pause this video and see,

• first of all, does this theorem apply?

• And if it does apply, what is this limit?

• So the first thing we need to see

• is does this theorem apply?

• So first of all, if we were to find the limit

• as x approaches negative three of g of x, what is that?

• Well, when we're approaching negative three from the right,

• it looks like our function is actually at three

• and it looks like when we're approaching negative three

• from the left, it looks like our function is at three.

• So it looks like this limit is three,

• even though the value g of negative three is negative two,

• but it's a point discontinuity.

• As we approach it from either side,

• the value of the function is at three.

• So this thing is going to be three,

• so it exists, so we meet that first condition

• and then the second question is is our function f

• continuous at this limit, continuous at three?

• So when x equals three, yeah, it looks like at that point,

• our function is definitely continuous

• and so we could say that this limit

• is going to be the same thing

• as this equals f of the limit

• as x approaches negative three of g of x,

• close the parentheses

• and we know that this is equal to three

• and we know that f of three

• is going to be equal to negative one.

• So this met the conditions for this theorem

• and we were able to use the theorem

• to actually solve this limit.

- [Tutor] In this video, we're going to try to understand

A2 初級 美國腔

# 1.44 複合函數極限定理（Theorem for limits of composite functions | Limits and contiuity | AP Calculus | Khan Academy）

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yukang920108 發佈於 2022 年 07 月 01 日