  In this paper we give a representation formula for the limit of the fnite
horizon problem as the horizon becomes infinite, with a nonnegative Lagrangian
and unbounded data. It is related to the limit of the discounted infinite
horizon problem, as the discount factor goes to zero. We give sufficient
conditions to characterize the limit function as unique nonnegative solution of
the associated HJB equation. We also briefly discuss the ergodic problem.
