  In this paper we study a non-homogeneous Neumann problem, where the
$p(x)$-Laplacian is involved and $p=\infty$ in a subdomain. By considering a
suitable sequence $p_k$ of bounded variable exponents such that $p_k \to p$ and
replacing $p$ with $p_k$ in the original problem, we prove the existence of a
solution $u_k$ for each of those intermediate ones. We show that the limit of
the $u_k$ exists and after giving a variational characterization of it, in the
part of the domain where $p$ is bounded, we show that it is a viscosity
solution in the part where $p=\infty$. Finally, we formulate the problem of
which this limit function is a solution in the viscosity sense.
