Asymptotics of the $s$-perimeter as $s\searrow 0$

We deal with the asymptotic behavior of the $s$-perimeter of a set $E$ inside a domain $\mathrm{\Omega }$ as $s\searrow 0$. We prove necessary and sufficient conditions for the existence of such limit, by also providing an explicit formulation in terms of the Lebesgue measure of $E$ and $\mathrm{\Omega }$. Moreover, we construct examples of sets for which the limit does not exist.