I need to find a constant $\alpha$ and a sequence of constants $\beta_{1}, \beta_{2}, \ldots$ such that $n^{\alpha}\left(X_{(n)}-\beta\right)$ has a non-degenerate limiting distribution.
Here $X_{1}, \ldots, X_{n}$ are i.i.d. random vectors such that $X_{i}$ has an absolutely continuous distribution with density function $$ \frac{\exp (-x)}{[1+\exp (-x)]^{2}}, x \in \mathbb{R}. $$ And $X_{(n)}=\max (X_{i}) .$