The main aim of this technical note is to present a new approach to studying the asymptotic behavior of nonlinear delay differential systems. Firstly, we develop a new nonlinear delay differential inequality which will be more effective and interesting than earlier ones in [1]-[4]. Then we establish a new generalized Barbalat's lemma which can extend and improve earlier ones in [5]-[8]. Finally, by combining the obtained generalized Barbalat's lemma with the nonlinear delay differential inequality, we obtain the attracting set and invariant set of the nonlinear delay differential systems.