Find $\lim_{n\rightarrow\infty} e^{n+\frac{1}{n}}-e^n$? $\lim_{n\rightarrow\infty} e^{n+\frac{1}{n}}-e^n$
I tried multiplying and dividing with $e^{n+\frac{1}{n}}+e^n$ and then doing l'hopital, but I got nowhere

$e^{n+1/n}-e^n=e^n(e^{1/n}-1)={e^n\over n}{{e^{1/n}-1}\over {1\over n}}$. You have
$lim_{n\rightarrow +\infty}{{e^{1/n}-1}\over {1\over n}}$ is the derivative of $exp$ at $0$ so $is$ $1$ and $lim_{n\rightarrow +\infty}{e^n\over n}=+\infty$.
