Using a modification of the Tawumikhin method, the authors obtain asymptotic properties of solutions os delay differential equations. In particular using Liapunov functions we obtain sufficient conditions for solutions to approach a constant c as t→∞. Here and f has appropriate smoothness properties to guarantee extendability of solutions. These results are applied to the widely studied class of delay equations , where are continuous γ is ratio of odd integers, e>0, and r>0. When c=0 new results on asymptotic stability are obtained.