Prove Power Rule for Limits: $\lim_{x \to a} f(x)^{g(x)} = \left(\lim_{x \to a} f(x)\right)^{\lim_{x\to a} g(x)}$

Question

Suppose

• $\lim_{x \to a} f(x) = L$
• $\lim_{x \to a} g(x) = M$

I would like to prove

$$\lim_{x \to a} f(x)^{g(x)} = \left(\lim_{x \to a} f(x)\right)^{\lim_{x\to a} g(x)}=L^M$$

I thought this had probably been asked before, but the closest I can find is Prove $\lim_{x \to a} f(x)^n = \left(\lim_{x \to a} f(x)\right)^n$.

All limit law proofs I found on Google excluded this one. However it seems to be true, as it is listed on this Wikipedia article (sadly without proof).

I imagine the proof is quite difficult, since $g$ can take on integer, rational, and irrational values in general, all of which would change the meaning of $f^g$.

Thanks in advance.