TITLE: How to evaluate the limit where something is raised to a power of $x$?
QUESTION [9 upvotes]: I am attempting to evaluate the following limit:
$$\lim_{x\to \infty} \Biggl(\frac{x+3}{x+8}\Biggl)^x$$
I was wondering if anyone could share some strategies for evaluating limits raised to a power of $x$, as I have never encountered these before.
I have found the answer to be $\frac{1}{e^5}$, but I am unsure how to arrive at this answer.

REPLY [3 votes]: $$\begin{align}
\lim_{x\to\infty}\left(\frac{x+3}{x+8}\right)^x&=\lim_{x\to\infty}\left(1-\frac5{x+8}\right)^{x+8}\left(1-\frac5{x+8}\right)^{-8}\\
&=e^{-5}
\end{align}$$