# Evaluating the limit $\lim_{n \rightarrow +\infty} \frac{e^n+e^{-n}}{e^{n+1}+e^{-n-1}}$

How would you solve the following limit? It's $\frac \infty \infty$ and L'Hospital doesn't seem to help:

$$\lim_{n \rightarrow +\infty} \frac{e^n+e^{-n}}{e^{n+1}+e^{-n-1}}$$

Hint: Multiply the numerator and denominator by $e^{-n}$.
HINT: Divide the numerator and the denominator by $\mathrm{e}^n$: $$\frac{e^n+e^{-n}}{e^{n+1}+e^{-n-1}} = \frac{1+e^{-2n}}{e+e^{-2n-1}}$$
Intuition: As $n$ gets very large, $e^{-n}$ and $e^{-n-1}$ both get very, very small, so what you're left with should be $$\lim_{n\to\infty} \frac{e^n}{e^{n+1}}.$$