Summary: 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 \(\lim_{\alpha\to 0}\alpha u_\alpha(\theta,x) = \xi(\theta)\) where \(\xi(\theta)\) is the average on \(]0,\theta[\) of the optimal cost of the (usual) infinite horizon risk-sensitive control problem.