This article presents the exact limiting value of the derivative of the excess Helmholtz energy, A(E), with respect to molecular size at constant temperature, density and composition for a binary mixture of hard spheres with an infinite size ratio (sigma(11)/sigma(22)-->infinity); i.e., lim(sigma 22-->0)[(partial derivative A(hs)(E)/RT)/partial derivative sigma(22)](T),(rho,x,sigma 11) = (pi/W)rho x(1)x(2)sigma(11)(2)/(1-(pi/6)rho x(1)sigma(11)(3)). This limiting value is compared with the Mansoori-Carnahan-Starling-Leland (MCSL) and also used to test the limits of some commonly used models in estimating the excess free energy of solvents in mixtures or polymer solutions. The models evaluated include the van Laar, Wilson, Edmond-Ogston, Flory-Huggins, Lacome-Sanchez, Scott-Magat, and Chen et al. It is shown that while the MCSL equation of state produces the same limiting value as the exact value reported here the other mixture models deviate from the exact value. This expression may be utilized to correct the mixture theories at their infinite size ratio limits.