We consider the discounted approximation of the critical Hamilton-Jacobi equation set on the real line associated with the Hamiltonian G(x,p):=H(x,p)-V(x), where H is a 1-periodic Tonelli Hamiltonian and V is a continuous and compactly supported potential. The critical constant associated with G is characterized as the unique constant a for which the associated HJ equation G(x,u')=a can have globally bounded solutions. We prove that the solutions of the discounted equation converge to a specific critical solution, which is identified in terms of projected Mather measures for G and of the asymptotic solution of the unperturbed periodic problem. This is joint work with I. Capuzzo-Dolcetta.