Placeholder Image

字幕列表 影片播放

  • - [Instructor] We are asked to find

  • these three different limits.

  • I encourage you like always,

  • pause this video and try to do it yourself

  • before we do it together.

  • So when you do this first one,

  • you might just try to find the limit

  • as x approaches negative two of f of x

  • and then the limit as x approaches negative two of g of x

  • and then add those two limits together.

  • But you will quickly find a problem,

  • 'cause when you find the limit

  • as x approaches negative two of f of x,

  • it looks as we are approaching negative two

  • from the left, it looks like we're approaching one.

  • As we approach x equals negative two from the right,

  • it looks like we're approaching three.

  • So it looks like the limit

  • as x approaches negative two of f of x doesn't exist,

  • and the same thing's true of g of x.

  • If we approach from the left,

  • it looks like we're approaching three.

  • If we approach from the right,

  • it looks like we're approaching one.

  • But it turns out that this limit can still exist

  • as long as the limit as x approaches negative two

  • from the left of the sum,

  • f of x plus g of x,

  • exists and is equal to the limit

  • as x approaches negative two from the right of the sum,

  • f of x plus g of x.

  • So what are these things?

  • Well, as we approach negative two from the left,

  • f of x is approaching, looks like one,

  • and g of x is approaching three.

  • So it looks like we're approaching one and three.

  • So it looks like this is approaching.

  • The sum is going to approach four.

  • And if we're coming from the right,

  • f of x looks like it's approaching three

  • and g of x looks like it is approaching one.

  • Once again, this is equal to four.

  • And since the left and right handed limits

  • are approaching the same thing,

  • we would say that this limit exists and it is equal to four.

  • Now let's do this next example as x approaches one.

  • Well, we'll do the exact same exercise.

  • And once again, if you look at the individual limits

  • for f of x from the left and the right as we approach one,

  • this limit doesn't exist.

  • But the limit as x approaches one of the sum might exist,

  • so let's try that out.

  • So the limit as x approaches one

  • from the left hand side of f of x plus g of x,

  • what is that going to be equal to?

  • So f of x, as we approach one from the left,

  • looks like this is approaching two.

  • I'm just doing this for shorthand.

  • And g of x, as we approach one from the left,

  • it looks like it is approaching zero.

  • So this will be approaching two plus zero, which is two.

  • And then the limit,

  • as x approaches one from the right hand side

  • of f of x plus g of x is going to be equal to.

  • Well, for f of x as we're approaching one

  • from the right hand side,

  • looks like it's approaching negative one.

  • And for g of x as we're approaching one

  • from the the right hand side,

  • looks like we're approaching zero again.

  • Here it looks like we're approaching negative one.

  • So the left and right hand limits

  • aren't approaching the same value,

  • so this one does not exist.

  • And then last but not least,

  • x approaches one of f of x times g of x.

  • So we'll do the same drill.

  • Limit as x approaches one from the left hand side

  • of f of x times g of x.

  • Well, here, and we can even use the values here.

  • We see it was approaching one from the left.

  • We are approaching two, so this is two.

  • And when we're approaching one from the left here,

  • we're approaching zero.

  • We're gonna be approaching two times zero, which is zero.

  • And then we approach from the right.

  • X approaches one from the right

  • of f of x times g of x.

  • Well, we already saw when we're approaching one

  • from the right of f of x,

  • we're approaching negative one.

  • But g of x, approaching one from the right,

  • is still approaching zero,

  • so this is going to be zero again, so this limit exists.

  • We get the same limit

  • when we approach from the left and the right.

  • It is equal to zero.

  • So these are pretty interesting examples,

  • because sometimes when you think

  • that the component limits don't exist

  • that that means that the sum

  • or the product might not exist,

  • but this shows at least two examples

  • where that is not the case.

- [Instructor] We are asked to find

字幕與單字

單字即點即查 點擊單字可以查詢單字解釋

A2 初級 美國腔

1.43分段函數(Limits of combined functions: piecewise functions | AP Calculus AB | Khan Academy)

  • 0 1
    yukang920108 發佈於 2022 年 07 月 01 日
影片單字