Raising each side of a limit to a power? If you have $\lim_{x\to \infty} f(x)=L$ $\quad$(or as $x \rightarrow c$ for that matter), is it correct to say that $\lim_{x\to \infty} f(x)^{g(x)}=L^{\lim_{x\to \infty}g(x)}$ as long as $L \not= 1$? I'm not looking for a very rigorous response, I am just looking for if this is a correct technique in general that can help you evaluate a limit. Thanks.
 A: I followed the comment discussion where the question arose. 
I will give a sequence of more or less trivial cases.
The most easiest case is to ask whether 
$$ \lim_{n\to \infty} f(x_n)=f(\lim_{n\to \infty} x_n)$$
holds. This is always the case when $f$ is continuous and might be true even if $f$ is not continuous. 
the next case is to ask whether 
$$ \lim_{n\to \infty} f_n(x_n)= \lim_{k\to \infty} f_k (\lim_{n\to \infty }(x_n))$$
holds. Here we need more than just every $f_k$ to be continous, we even need that the 
limit function ist continuous. This is always the case if $f_n$ converges uniformly. 
In your case as 
$$ f(x)^{g(x)} = \exp(\log(f(x))\cdot g(x))$$
Here you see that the case where $f(x)$ goes to $1$ surely could cause trouble. Furthermore 
negative values of $f(x)$ has to be handled in a different way. 
In the case where you question comes from you looked at the sequence of functions $x^n$. This converges pointwise on $[0,1]$ diverges everywhere else and converges uniform due to Dini's Theorem on $[0,1-\varepsilon]$. So whenever the limit is strictly lower than $1$ you may do it. 
