Limit of the normalised maximum of exponential variables

Let $X_1,…,X_n$ be i.i.d. random variables with the probability density function

$$f(x)=\begin{cases}e^{-x}, & x>0,\ 0, & \text{otherwise.}\end{cases}$$

If $X_{(n)}=\max\{X_1,…,X_n\}$, then $\lim_{n\to\infty}P\left(X_{(n)}-\log_en\le2\right)$ equals