  We study the asymptotic behaviour of suitably defined seminorms in general
metric measure spaces. As a particular case we provide new and shorter proofs
of the Maz'ya-Shaposhnikova's theorem on the asymptotic behaviour of the
fractional Sobolev $s$-seminorm, in the setting of metric measure spaces and
with general mollifiers, as well as of the Ludwig's result on finite
dimensional Banach spaces. Our result also provides new spaces satisfying an
asymptotic formula and it also builds a link between the asymptotic formula for
functions and the asymptotic volume ratio of a metric measure space. In
addition, we prove two related rigidity results for metric measure spaces with
synthetic Ricci curvature bound which are new even in the smooth setting.
