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

Notice that $$\left(\frac{f(x)}{f(x)+1} \right)^{f(x)} = \frac{1}{\left(1+\frac{1}{f(x)} \right)^{f(x)}}$$
