As part of a larger problem (showing convergence in distribution of a random variable), I am trying to find $\displaystyle{\lim_{n\to\infty}} \sqrt n \left(\log\left(\frac x{e\sqrt n} + 1\right)\right)$. I know that the answer is $\frac xe$ (via WolframAlpha) but I don't know how to show it.
So far I have tried manipulating the expression to make it amenable to L'Hopital's rule, but this hasn't succeeded; I wind up just getting another indeterminate expression after applying L'Hopital's. Any tips would be appreciated.