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 $g$.

We have that 
$$(1+f(n))^{g(n)}=e^{g(n)\log(1+f(n))}$$
and since
$$g(n)\log(1+f(n))=g(n)\cdot f(n)\cdot \frac{\log(1+f(n))}{f(n)}$$
with
$$\frac{\log(1+f(n))}{f(n)}\to 1$$
all boils down in $g(n)f(n)$.
