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