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.