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  • - [Instructor] All right, let's get

  • a little bit more practice taking limits

  • of composite functions.

  • Here, we want to figure out what is the limit

  • as x approaches negative one of g of h of x?

  • The function g, we see it defined graphically

  • here on the left, and the function h,

  • we see it defined graphically here on the right.

  • Pause this video and have a go at this.

  • All right, now your first temptation might be to say,

  • all right, what is the limit as x approaches negative one

  • of h of x, and if that limit exists, then input that into g.

  • If you take the limit as x approaches negative one

  • of h of x, you see that you have a different limit

  • as you approach from the right

  • than when you approach from the left.

  • So your temptation might be to give up at this point,

  • but what we'll do in this video is to realize

  • that this composite limit actually exists

  • even though the limit as x approaches negative one

  • of h of x does not exist.

  • How do we figure this out?

  • Well, what we could do is take right-handed

  • and left-handed limits.

  • Let's first figure out what is the limit

  • as x approaches negative one from the right hand side

  • of g of h of x?

  • Well, to think about that, what is the limit of h

  • as x approaches negative one from the right hand side?

  • As we approach negative one from the right hand side,

  • it looks like h is approaching negative two.

  • Another way to think about it is this is going to be

  • equal to the limit as h of x approaches negative two,

  • and what direction is it approaching negative two from?

  • Well, it's approaching negative two from values

  • larger than negative two.

  • H of x is decreasing down to negative two

  • as x approaches negative one from the right.

  • So it's approaching from values larger than negative two

  • of g of h of x.

  • G of h of x.

  • I'm color coding it to be able to keep track of things.

  • This is analogous to saying what is the limit,

  • if you think about it as x approaches negative two

  • from the positive direction of g?

  • Here, h is just the input into g.

  • So the input into g is approaching negative two

  • from above, from the right I should say,

  • from values larger than negative two,

  • and we can see that g is approaching three.

  • So this right over here is going to be equal to three.

  • Now, let's take the limit as x approaches negative one

  • from the left of g of h of x.

  • What we could do is first think about what is h approaching

  • as x approaches negative one from the left?

  • As x approaches negative one from the left,

  • it looks like h is approaching negative three.

  • We could say this is the limit

  • as h of x is approaching negative three,

  • and it is approaching negative three

  • from values greater than negative three.

  • H of x is approaching negative three from above,

  • or we could say from values greater than negative three,

  • and then of g of h of x.

  • Another way to think about it,

  • what is the limit as the input to g

  • approaches negative three from the right?

  • As we approach negative three from the right,

  • g is right here at three,

  • so this is going to be equal to three again.

  • So notice the right hand limit and the left hand limit

  • in this case are both equal to three.

  • So when the right hand and the left hand limit

  • is equal to the same thing, we know that the limit

  • is equal to that thing.

  • This is a pretty cool example,

  • because the limit of, you could say the internal function

  • right over here of h of x, did not exist,

  • but the limit of the composite function still exists.

- [Instructor] All right, let's get

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A2 初級 美國腔

1.46 複合函數的極限:不存在內部極限(Limits of composite functions: internal limit doesn't exist | AP Calculus | Khan Academy)

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    yukang920108 發佈於 2022 年 07 月 01 日
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