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  • - [Instructor] Let's think a little bit about

  • limits of piecewise functions that are defined algebraically

  • like our f of x right over here.

  • Pause this video and see if you can figure out

  • what these various limits would be,

  • some of them are one-sided,

  • and some of them are regular limits, or two-sided limits.

  • Alright, let's start with this first one,

  • the limit as x approaches four,

  • from values larger than equaling four,

  • so that's what that plus tells us.

  • And so when x is greater than four,

  • our f of x is equal to square root of x.

  • So as we are approaching four from the right,

  • we are really thinking about this part of the function.

  • And so this is going to be equal to the square root of

  • four, even though right at four,

  • our f of x is equal to this,

  • we are approaching from values greater than four,

  • we're approaching from the right, so we would use

  • this part of our function definition,

  • and so this is going to be equal to two.

  • Now what about our limit of f of x,

  • as we approach four from the left?

  • Well then we would use this part of our function definition.

  • And so this is going to be equal to four plus two

  • over four minus one,

  • which is equal to 6 over three,

  • which is equal to two.

  • And so if we wanna say what is the limit of f of x

  • as x approaches four, well this is a good scenario here

  • because from both the left and the right

  • as we approach x equals four, we're approaching

  • the same value, and we know, that in order for

  • the two side limit to have a limit, you have to be

  • approaching the same thing from the right and the left.

  • And we are, and so this is going to be equal to two.

  • Now what's the limit as x approaches two of f of x?

  • Well, as x approaches two, we are going to be

  • completely in this scenario right over here.

  • Now interesting things do happen at x equals one here,

  • our denominator goes to zero, but at x equals two,

  • this part of the curve is gonna be continuous

  • so we can just substitute the value, it's going to be

  • two plus two, over two minus one, which is four over one,

  • which is equal to four.

  • Let's do another example.

  • So we have another piecewise function,

  • and so let's pause our video and figure out these things.

  • Alright, now let's do this together.

  • So what's the limit as x approaches negative one

  • from the right?

  • So if we're approaching from the right,

  • when we are greater than or equal to negative one,

  • we are in this part of our piecewise function,

  • and so we would say, this is going to approach,

  • this is gonna be two, to the negative one power,

  • which is equal to one half.

  • What about if we're approaching from the left?

  • Well, if we're approaching from the left,

  • we're in this scenario right over here,

  • we're to the left of x equals negative one,

  • and so this is going to be equal to the sine,

  • 'cause we're in this case, for our piecewise function,

  • of negative one plus one, which is the sine of zero,

  • which is equal to zero.

  • Now what's the two-sided limit as x approaches

  • negative one of g of x?

  • Well we're approaching two different values

  • as we approach from the right,

  • and as we approach from the left.

  • And if our one-sided limits aren't approaching

  • the same value, well then this limit does not exist.

  • Does not exist.

  • And what's the limit of g of x,

  • as x approaches zero from the right?

  • Well, if we're talking about approaching zero

  • from the right, we are going to be in this case

  • right over here, zero is definitely in this interval,

  • and over this interval, this right over here

  • is going to be continuous, and so we can just substitute

  • x equals zero there, so it's gonna be two to the zero,

  • which is, indeed, equal to one, and we're done.

- [Instructor] Let's think a little bit about

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A2 初級 美國腔

1.54 分段函數的極限(Limits of piecewise functions | Limits and continuity | AP Calculus AB | Khan Academy)

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    yukang920108 發佈於 2022 年 07 月 02 日
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