  We deal with the asymptotic behavior of the $s$-perimeter of a set $E$ inside
a domain $\Omega$ as $s\searrow0$. 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 $\Omega$. Moreover, we construct
examples of sets for which the limit does not exist.
