We consider the infinite frequency moduli and time correlation functions of fluids composed of particles that interact through a steeply repulsive potential of the general analytic form, phi(r)=k(B)T exp[-alphaf(r)], where alpha is a measure of the steepness or stiffness of the potential. Although these potentials have different analytical forms, in the steeply repulsive limit of alpha-->infinity, the derived properties become almost identical and are only dependent on the value of alpha and other basic variables. All the infinite frequency moduli which we study are proportional to alpha and the interaction part of the pressure is only weakly dependent on alpha. For the force and other configurational property time functions C(t), time t can be replaced by alphat, i.e., C(t)=1-T*(alphat*)(2)+O[(alphat*)(4)], where T*=k(B)T/epsilon, is the reduced temperature, k(B) is Boltzmann's constant, where epsilon is a characteristic energy for the potential, and t* is a reduced time. We proved this in earlier publications for an inverse power, r(-n) potential (where alpha=n), and show here this more general relationship. The effective hard-sphere diameter by the Barker-Henderson equation, and an alternative prescription derived here, give to first order in alpha(-1) the same formula for the effective hard-sphere diameter for these potentials. We have carried out molecular-dynamics simulations that confirm the equivalence in the steeply repulsive limit of both the static and dynamical properties of two such potentials, which have an inverse power r(-n) and exponential potential exp(-kappar) form. We consider that the theory for the infinite frequency shear rigidity modulus presented here could be usefully applied to predict the infinite frequency storage modulus of colloidal liquids. (C) 2003 American Institute of Physics.