# Limit problem

## Main Question or Discussion Point

I have a question about the limit problem. I need find limit if it exists.

lim x (approach to 4) (3x+square root x-4).
If I plug number for x=4, I get 12. But I am concern about square root. Do I need find one-sided limit from x approach to 4+ and x approach to 4-. Or 12 will be the correct answer?
Thanks, everyone

dextercioby
Homework Helper
If it's not specified,then the superior limit is understood.

Daniel.

mathman
Usually the limit (in the case of real variables) is defined by getting limits on both sides. If they are equal, that is the limit. If not, the limit does not exist.

HallsofIvy
Homework Helper
Except when it is clear that the function itself does not exist on one side. Then the one sided limit is the limit.

dextercioby
Homework Helper
We both assumed the limit & the function were defined on the reals (real intervals).

Daniel.

Limit Laminant...

$$\boxed{\lim_{x \rightarrow 4} 3x + \sqrt{x - 4} = 12}$$

Hurkyl
Staff Emeritus
Gold Member
You should ask your professor how he would like you to answer. However, I think you should take the strict interpretation -- it's written as a two-sided limit, so don't assume it's a one-sided limit.

When writing your answer, you can always say the limit doesn't exist, and then write down what the one-sided limit is.

P.S. dex: I know everyone knows what you meant, but "limit superior" (written "lim sup") means something else.

Yes, I would use the definition. By the definition, if you're working over the reals, the limit doesn't exist. As Hurkyl said, putting down the one-sided limit as well couldn't hurt. But unless it has been clearly stated in class that you have a different definition for limits when the function doesn't exist on one side, then you shouldn't assume so.

Thanks guys a lot