Prove general operation on functions goes to one

Question

Given two functions $f(x)$ and $g(x)$, where it's known that $\lim_{x\to 0} f(x)=0$ and $\lim_{x\to 0}g(x)=0$, is it possible to prove that $\lim_{x\to 0} f(x)^{g(x)}=1$? If it's not possible to prove this, what's a counter-example to this claim?

Answer 1

Let $f(x)=e^{-1/x^2}$ and $g(x)=x$. Both these functions have a limit of zero as $x\to0$, but $$\lim_{x\to0}f(x)^{g(x)}=\lim_{x\to0}e^{-1/x}$$ does not even exist (the right-hand side limit is zero and the expression diverges to $+\infty$ from the left-hand side).