A class of singularly perturbed initial-and boundary-value problems occurring in discrete systems is formulated. Singular perturbation methods are developed separately for initial-value problems and boundary-value problems. The methods essentially consist of seeking an approximate solution in terms of an outer series basod on the degenerate problem and a correction series to recover the auxiliary conditions sacrificed in the process of degeneration. Some interesting features are explorod and the asymptotic correctness of the formal series expansions is clearly established. Illustrative examples are provided in support of the proposed methods.