Name: Limits of piecewise functions | Limits and continuity | AP Calculus AB | Khan Academy
Uploaded: 2022-07-02T03:00:09.000Z
Duration: 3 min 49 s
Description: 分段函數的極限 |限制和連續性 | AP微積分AB |可汗學院Learn the English Usage: equal,approaching,limit,sided,negative,approach,scenario,square root,interval,greater,sine,substitute,root,continuous,pause,definition,square,exist,figure,denominator,case,continuity,curve,instructor,calculus,defined,khan...

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

limits of piecewise functions that are defined algebraically

Pause this video and see if you can figure out

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

Alright, let's start with this first one,

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

we are approaching from values greater than four,

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

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

And so this is going to be equal to four plus 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

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,

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

So what's the limit as x approaches negative one

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,

What about if we're approaching from the left?

Well, if we're approaching from the left,

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,

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

Well we're approaching two different values

And if our one-sided limits aren't approaching

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

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.