Limit Help: $\lim_{x\to\infty} xe^{-a\frac{x}{\ln x}}$ I feel dumb for asking this, but I couldn't quite show that this limit is 0 (which I think is correct) whenever $a>0$: $$\lim_{x\to\infty} xe^{-a\frac{x}{\ln x}}...--prophetes.ai

Limit Help: $\lim_{x\to\infty} xe^{-a\frac{x}{\ln x}}$ I feel dumb for asking this, but I couldn't quite show that this limit is 0 (which I think is correct) whenever $a>0$: $$\lim_{x\to\infty} xe^{-a\frac{x}{\ln x}}.$$ I tried using L'Hospital's rule which was mildly annoying given the amount of terms. I also couldn't figure out a good way to use comparison either.

Hint: try to use the fact that $\log x < \sqrt{x}$ for $x \gg 1$