Suppose $\lim _{x\to \infty} f(x) = \infty$. Calculate $\lim_{x\rightarrow \infty} \left(\frac{f(x)}{f(x)+1}\right)^{f(x)}$ Suppose $\lim _{x\to \infty} f(x) = \infty$. Calculate: $\lim_{x\rightarrow 
 \infty} \left(\dfrac{f(x)}{f(x)+1}\right)^{f(x)}$
I figured the limit is $\dfrac{1}{e}$, but I have to prove it using the definition of limit, not sure how. Thanks

Since $f(x)\to\infty$ as $x\to\infty$, we can think of the limit as one where $n=f(x)$ and $n\to\infty$. Hence, $$\lim_{x\to\infty}\left(\frac{f(x)}{1+f(x)}\right)^{f(x)}=\lim_{n\to\infty}\left(\frac{n}{1+n}\right)^n=\lim_{n\to\infty}\left(\frac{1+n}{n}\right)^{-n}=\left[\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n\right]^{-1}$$
