The limit as epsilon --> 0 of the value function of a singularly perturbed optimal control problem is characterized. Under general conditions it is shown that limit value functions exist and solve in a viscosity sense a Hamilton-Jacobi equation. The Hamiltonian of this equation is generated by an infinite horizon optimization on the fast time scale. In particular, the limit Hamiltonian and the limit Hamilton-Jacobi equation are applicable in cases where the reduction of order, namely setting epsilon = 0, does not yield an optimal behavior.