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  • - [Instructor] Let's find the limit of f of x

  • times h of x as x approaches zero.

  • All right, we have graphical depictions

  • of the graphs y equals f of x and y equals h of x.

  • And we know, from our limit properties,

  • that this is going to be the same thing as the limit

  • as x approaches zero of f of x

  • times,

  • times the limit

  • as x approaches zero of

  • h of x.

  • And let's think about what each of these are.

  • So let's first think about f of x right over here.

  • So on f of x, as x approaches zero,

  • notice the function itself isn't defined there.

  • But we see when we approach from the left,

  • we are approaching the,

  • the function seems to be approaching the value

  • of negative one right over here.

  • And as we approach from the right, the function seems

  • to be approaching the value of negative one.

  • So the limit here,

  • this limit here is negative one.

  • As we approach from the left,

  • we're approaching negative one.

  • As we approach from the right, the value of the function

  • seems to be approaching negative one.

  • Now what about h of x?

  • Well, h of x we have down here.

  • As x approaches zero,

  • as x approaches zero,

  • the function is defined at x equals zero.

  • It looks like it is equal to one.

  • And the limit is also equal to one.

  • We can see that, as we approach it from the left,

  • we are approaching one.

  • As we approach from the right, we're approaching one.

  • As we approach x equals zero from the left, we approach,

  • the function approaches one.

  • As we approach x equals zero from the right,

  • the function itself is approaching one.

  • And it makes sense that the function is defined there,

  • is defined at x equals zero,

  • and the limit as x approaches zero

  • is equal to the same as the,

  • is equal to the value of the function at that point

  • because this is a continuous function.

  • So this is,

  • this is one.

  • And so negative one times one

  • is going to be equal to,

  • is equal to negative one.

  • So that is equal to negative one.

  • Let's do one more.

  • All right, so these are both,

  • looks like continuous functions.

  • So we have the limit as x approaches zero of h of x

  • over g of x.

  • So once again, using our limit properties,

  • this is going to be the same thing as the limit of h of x

  • as x approaches zero

  • over the limit

  • of g of x

  • as x approaches zero.

  • Now what's the limit of h of x as x approaches zero?

  • This is, let's see,

  • as we approach zero from the left,

  • as we approach x equals zero from the left,

  • our function seems to be approaching four.

  • And as we approach x equals zero from the right,

  • our function seems to be approaching four.

  • That's also what the value

  • of the function is at x equals zero.

  • That makes sense because this is a continuous function.

  • So the limit as we approach x equals zero should be the same

  • as the value of the function at x equals zero.

  • So this top, this is going to be four.

  • Now let's think about the limit of g of x as x equals,

  • as x approaches zero.

  • So from the left, it looks like, as x approaches zero,

  • the value of the function is approaching zero.

  • And as x approaches zero from the right,

  • the value of the function is also approaching zero,

  • which happens to also be,

  • which also happens to be g of zero.

  • G of zero is also zero.

  • And that makes sense that the limit and the actual value

  • of the function at that point is the same

  • because it's continuous.

  • So this also is zero,

  • but now we're in a strange situation.

  • We have to take four and divide it by zero.

  • So this limit will not exist

  • 'cause we can't take four and divide it by zero.

  • So even though the limit of h of x is x equals,

  • as x approaches zero exists

  • and the limit of g of x as x approaches zero exists,

  • we can't divide four by zero,

  • so this whole entire limit does not exist,

  • does not exist.

  • And actually, if you were to plot h of x over g of x,

  • if you were to plot that graph,

  • you would see it even clearer

  • that that limit does not exist.

  • You would actually be able to see it graphically.

- [Instructor] Let's find the limit of f of x

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A2 初級 美國腔

1.42 組合函數的極限(Limits of combined functions | Limits and continuity | AP Calculus AB | Khan Academy)

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    yukang920108 發佈於 2022 年 07 月 01 日
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