Hello;

What is the limit of $\displaystyle \frac{({\log{2}^n})^{\log{2}^n}}{({\log{2}^{n+1})} ^{\log{2}^{n+1}}}$ as $\displaystyle n$ tends to infinity? To my mind, it's either $\displaystyle \frac{1}{e}$ or $\displaystyle 0$. And I'm leaning towards $\displaystyle \frac{1}{e}$, but I'm not sure.

Thank you very much.