# Limit-Fundamental Concept?

Can anyone clear me this fundamental concept about this. I am confused for many months over this.

It is said that in $\lim_{x\to a} f(x)^{g(x)}$.
$f(x)$ should be greater then $0$.
Can anyone explain the reason?

If $f(x)$ is $-2$ and $g(x)$ is $\frac{1}{2}$,then it is not possible. But if $f(x)$ is $-2$ and $g(x)$ is $2$,then why it is not possible? (It is true as $(-2)^2 = 4$)

As it has been pointed out, $\lim_{x \to a} (-2)^{1/2} = i\sqrt{2}$ so using complex numbers changes this. But if you haven't learned about complex numbers, it often said that $f(x)$ in $\lim_{x\to a}f(x)^{g(x)}$ has to be positive, mostly because it is easier to do that way. As you have shown there are cases where the limit exists even though $f(x)$ isn't positive, but they are few, and uncommon enough (at least until you have reached a level where you will have learned about complex numbers) that they can dealt with.
And don't forget that in taking the limit you have specified $g(x)$ varies as well as $f(x)$ so you are potentially dealing with a continuously changing real exponent rather than some nice integer or rational one - so you really need the exponential function to be valid for all reals, and continuous in the exponent. If $f(x)$ is positive, there isn't a problem.