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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.

• 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,

• 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

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