We study the Dirichlet problem for the p(x)-Laplacian, in the case when the variable exponent p(x) is infinite in a subdomain D. The main issue is to give a proper sense to what a solution is. To this end, we consider the limit of the solutions u_{n} to the corresponding problem when p_{n}(x) = min(p(x), n), in particular, with p_{n} = n in D. Under suitable assumptions on the data, we find that such a limit exists and that it can be characterized as the unique solution of a variational minimization problem. Moreover, we examine this limit in the viscosity sense and find an equation it satisfies.