  The conventional (Zwanzig-Mountain) expressions for instantaneous elastic
moduli of simple fluids predict their divergence as the limit of hard sphere
(HS) interaction is approached. However, elastic moduli of a true HS fluid are
finite. Here we demonstrate that this paradox reveals the soft- to hard-sphere
crossover in fluid excitations and thermodynamics. With extensive
\emph{in-silico} study of fluids with repulsive power-law interactions
($\propto r^{-n}$), we locate the crossover at $n\simeq 10-20$ and develop a
simple and accurate model for the HS regime. The results open novel prospects
to deal with the elasticity and related phenomena in various systems, from
simple fluids to melts and glasses.
