The F-signature is a fundamental numerical invariant of singularities in positive characteristic. Its positivity detects strong F-regularity, an important class of singularities related to KLT singularities in characteristic zero. In this article, we compute the limiting F-signature function of binomial and other related hypersurfaces in two variables as the characteristic
$p \to \infty $. In particular, we show it is a piecewise polynomial function, and relate it to the normalized volume.