The main purpose of this paper is to investigate the asymptotic behavior of the discounted risk-sensitive control problem for periodic diffusion processes when the discount factor alpha goes to zero. If u(alpha)(theta, x) denotes the optimal cost function, theta being the risk factor, then it is shown that lim(alpha -> 0)alpha u(alpha)(theta, x) = xi(theta) where xi(theta) is the average on]0, theta[ of the optimal cost of the (usual) infinite horizon risk-sensitive control problem.