In this paper, we investigate the problem of asymptotics of Lyapunov exponents for a class of singularly perturbed nonlinear systems. We define the maximal and minimal Lyapunov exponents for the underlying systems and show, via an averaging technique, that under certain conditions, the extremal Lyapunov exponents of the original system converge to the extremal Lyapunov exponents of the averaged slow subsystem when the singular perturbation parameter tends to zero. For low-dimensional systems, the existence of Lipschitz, continuous composite state feedbacks, which asymptotically provide the minimal Lyapunov exponents, can be shown. An example is given to illustrate the potential of the proposed technique and show that the designed controller is robust For sufficiently small perturbations.