For any positive integer $n\in\mathbb{N}$ let $S_n$ denote the set of all permutations (bijections) $\pi:\{1,\ldots,n\}\to \{1,\ldots,n\}$. For any $\pi\in S_n$ we let the *maximal displacement* be defined by $$\text{maxd}(\pi)= \max\big\{|k - \pi(k)|: k\in \{1,\ldots,n\}\big\}.$$
The expected value of the maximal displacements of all $\pi\in S_n$ is $$E^{\max}_n = \frac{1}{n!}\sum_{\pi\in S_n} \text{maxd}(\pi).$$

What is the value of $\lim_{n\to\infty} \frac{E^{\max}_n}{n}$?

(*Note.* The answer to this question seems to imply that $\lim_{n\to\infty} E^{\min}_n = 0$ if we define $E^{\min}_n$ in an analogous manner to $E^{\max}_n$ above, but I'm not sure this holds.)