Convergence of $(1+f(n))^{g(n)}$ when $f(n) \to 0$ and $g(n) \to +\infty$ Let $f : \Bbb N \to [0, \infty[$ and $g : \Bbb N \to [0, \infty[$ be two functions, such that $f(n) \to 0$ and $g(n) \to +\infty$ when $n \to \infty$.
What are some (necessary/sufficient) conditions on $f$ and $g$ for the limit
$$a_{f,g} := \lim_{n \to \infty} (1+f(n))^{g(n)}$$
to exist and be finite?
For instance, if $f(n) = x/n, g(n)=n$, then the limit is $e^x$. On the other hand, if $f(n)=1/n, g(n)=n^2$, then the limit is $+\infty$. 
If $f(n) = a^n, g(n)=n^k$ with $a \in [0,1[$ and $k \geq 1$, then the limit exists since $(1+a^n)^n$ is decreasing for $n$ large enough (take the log and then the derivative wrt $n$), and the sequence is bounded below by $1$. On the other hand, if $f(n) = a^n, g(n)=e^n$ (with $a \in [0,1[$) then the sequence $(1+f(n))^{g(n)}$ is unbounded.
Any help would be appreciated to understand better what conditions I should have on $f$ and $g$.

You have, since $\lim_{n\to\infty}f(n)=0$,
$$
(1+f(n))^{g(n)} = e^{g(n)\ln(1+f(n))} = \exp\left( {g(n)(f(n)+o(f(n)))} \right)
= \exp\left( {g(n)f(n)+o(f(n)g(n))} \right)
$$
This limit of $e^{a_n}$ exists if, and only if, the limit of $a_n$ exists in $\mathbb{R}\cup \{-\infty\}$.
so a necessary and sufficient condition is that $\lim_{n\to\infty} f(n)g(n)$ exists in $\mathbb{R}\cup \{-\infty\}$.
