We deal with the asymptotic behavior of the s-perimeter of a set E inside a domain U as s->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 U. Moreover, we construct examples of sets for which the limit does not exist. This is a joint work with A. Figalli, G. Palatucci, E. Valdinoci.