as the log of 3x when zero is less than x is less than three

and four minus x times the log of nine

when x is greater than or equal to three.

So based on this definition of g of x,

we want to find the limit of g of x

as x approaches three, and once again,

this three is right at the interface between

these two clauses or these two cases.

We go to this first case when x is between zero and three,

when it's greater than zero and less than three,

and then at three, we hit this case.

So in order to find the limit, we want to find

the limit from the left hand side

which will have us dealing with this situation

'cause if we're less than three we're in this clause,

and we also want to find a limit from the right hand side

which would put us in this clause right over here,

and then if both of those limits exist

and if they are the same, then that is going to be

the limit of this, so let's do that.

So let me first go from the left hand side.

So the limit as x approaches three from values

less than three, so we're gonna approach from the left

of g of x, well, this is equivalent to saying

this is the limit as x approaches three

from the negative side.

When x is less than three, which is what's happening here,

we're approaching three from the left,

we're in this clause right over here.

So we're gonna be operating right over there.

That is what g of x is when we are less than three.

So log of 3x,

and since this function right over here is defined

and continuous over the interval we care about,

it's defined continuous for all x's greater than zero,

we can just substitute three in here

to see what it would be approaching.

So this would be equal to log of three times three,

or logarithm of nine, and once again

when people just write log here within writing the base,

it's implied that it is 10 right over here.

So this is log base 10.

That's just a good thing to know

that sometimes gets missed a little bit.

All right, now let's think about the other case.

Let's think about the situation where we are

approaching three from the right hand side,

from values greater than three.

Well, we are now going to be in this scenario

right over there, so this is going to be equal

to the limit as x approaches three

from the positive direction, from the right hand side

of, well g of x is in this clause

when we are greater than three,

so four minus x times log of nine,

and this looks like some type of a logarithm expression

at first until you realize that log of nine

is just a constant, log base 10 of nine

is gonna be some number close to one.

This expression would actually define a line.

For x greater than or equal to three, g of x is just

a line even though it looks a little bit complicated.

And so this is actually defined for all real numbers,

and it's also continuous for any x that you put into it.

So to find this limit, to think about

what is this expression approaching

as we approach three from the positive direction,

well we can just evaluate a three.

So it's going to be four minus three

times log of nine, well that's just one,

so that's equal to log base 10 of nine.

So the limit from the left equals the limit from the right.

They're both log nine, so the answer here is

log log of nine,

and we are done.