TITLE: Proving that $\lim_{n\to\infty} \left(1+\frac{1}{f(n)}\right)^{g(n)} = 1$
QUESTION [8 upvotes]: 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.

REPLY [3 votes]: Note that
$$\lim_{n \to \infty}\frac{g(n)}{f(n)} = 0 \implies \bigg(1+\frac{1}{f(n)}\bigg)^{g(n)} = \Biggl[\bigg(1+\frac{1}{f(n)}\bigg)^{f(n)}\Biggl]^{\frac{g(n)}{f(n)}} = e^\frac{g(n)}{f(n)} \to  e^0 = 1$$