The dimensional analysis usually employed to solve the renormalization group equations for the asymptotic region is examined. It is argued that if this analysis is done systematically, one must in general add new inhomogeneous terms to the asymptotic equations even after invoking Weinberg's theorem to discard the (generalized) mass insertion term. These new irihomogeneities are entirely determined by the physical thresholds of the theory. They are shown to provide a natural explanation of Bjorken scaling in interacting field theories.