Originally Posted by
HallsofIvy First note that the discrete limit, $\displaystyle \lim_{n\to \infty} ne- n\left(\color{red}n+ \frac{1}{n}\right)^n$ is the same as the continuous limit $\displaystyle \large\lim_{x\to\infty} xe- x\left(\color{red}x+ \frac{1}{x}\right)^x$ as long as the latter limit exists. To do that, write it as $\displaystyle \large\lim_{x\to\infty}\frac{\color{red}2- \left(\color{red}x+ \frac{1}{x}\right)^x}{\frac{1}{x}}$, where both numerator go to 0, and use L'Hopital's rule.