calculus - Proving that $lim_{ntoinfty} left(1+frac{1}{f(n)}right)^{g(n)} = 1$

I want to prove that

$$\lim_{n\to\infty} \left(1+\frac{1}{f(n)}\right)^{g(n)} = 1$$ if $f(n)$ grows faster than $g(n)$ for $n\to\infty$ and $\lim_{n\to\infty} f(n) = +\infty = \lim_{n\to\infty}g(n)$.

It is quite easy to see that if $f = g$ the limit is $e$, but I can't find a good strategy to solve this problem.

Answer

We can use that

$$\left(1+\frac{1}{f(n)}\right)^{g(n)} =\left[\left(1+\frac{1}{f(n)}\right)^{f(n)}\right]^{\frac{g(n)}{f(n)}}$$