We know that $$ \lim_{n\to \infty} \left(1+\frac{1}{n}\right)^n=e $$ and $$ \lim_{n\to \infty} \left(1-\frac{1}{n}\right)^n=\frac{1}{e} $$
My question is: Are there any non-trivial:
- $f:\mathbb{N}\to\mathbb{R}$ monotonic non-decreasing
- $g:\mathbb{N}\to[-1,1]$
such that: $$ \lim_{n\to \infty} \left(1+g(n)\right)^{f(n)}=1 $$ It seems to me that except for the case where there exists an $N>0$ such that for all $n \ge N$ we have $g(n)=0$ - there are no other $f, g$ satisfying this limit.
Thank you.
Let $g(n)=1/n^2$, $f(n)=n$. Then $$(1+g(n))^{f(n)}=\left(1+\frac1{n^2}\right)^n=1+n\cdot\frac1{n^2}+\frac{n(n-1)}{2}\frac{1}{n^4}+\cdots\to 1.$$ In general, it suffices to have $g(n)$ going to zero sufficiently rapidly as $n\to\infty$.