We consider a collection of flows phi(epsilon) on R-n labeled by a parameter epsilon greater than or equal to 0. It is assumed that trajectories of phi(epsilon) converge - uniformly on compact time intervals - to trajectories of phi(0) as epsilon down arrow 0. Extra conditions are determined under which, if phi(0) has an asymptotically stable equilibrium point, then phi(epsilon) has an asymptotically stable equilibrium point. Examples illustrating the theory are included.