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

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

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Hint: Multiply the numerator and denominator by $e^{-n}$.

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  • $\begingroup$ oh.. I'm an idiot. Thank you $\endgroup$ – Crake Oct 27 '13 at 12:41
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HINT: Divide the numerator and the denominator by $\mathrm{e}^n$: $$ \frac{e^n+e^{-n}}{e^{n+1}+e^{-n-1}} = \frac{1+e^{-2n}}{e+e^{-2n-1}} $$

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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 $$\lim_{n\to\infty} \frac{e^n}{e^{n+1}}.$$

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