Using the exact solution valid for hard spheres in the proximity of infinite pressure, the correct limits for the compressibility (infinite), the residual Helmholtz energy (infinite), and the excess Helmholtz energy (finite) are derived. All the equations of state commonly used in chemical engineering applications fail to fulfil one of these limits, and are shown to be inconsistent in the limit of infinite pressure. (c) 2007 American Institute of Chemical Engineers.