We study the singular perturbation of optimal control problems for nonlinear systems with constraints on the fast state variables and a cost functional either of Bolza type or involving the exit time of the system from a given domain. Under a controllability assumption on the fast variables, we show that these variables become controls in the limit problem. Our method consists of passing to the limit in the associated Hamilton-Jacobi-Bellman (HJB) equations by means of some tools in the theory of viscosity solutions.