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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

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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
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