This paper investigates the limit behavior of a singularly perturbed control system with two state variables which are weakly coupled. The main novelty of our averaging approach lies in the fact that the limit dynamic may depend the initial condition of the fast system while in the literature this problem has been usually addressed under conditions that ensure that the limit dynamic is independent to this initial condition. Our study is based on a suitable nonexpansivity condition on the fast system which generalized dissipativity or stability properties of the fast dynamics.