  Let $\Omega\subset\mathbb{R}^{n}$ be a smooth bounded domain and $m\in
C(\overline{\Omega})$ be a sign-changing weight function. For $1<p<\infty$,
consider the eigenvalue problem $$
  \left\{ \begin{array} [c]{ll} -\Delta_{p}u=\lambda m(x)|u|^{p-2}u & \text{in
}\Omega,\\ u=0 & \text{on }\partial\Omega, \end{array} \right. $$ where
$\Delta_{p}u$ is the usual $p$-Laplacian. Our purpose in this article is to
study the limit as $p\rightarrow\infty$ for the eigenvalues $\lambda
_{k,p}\left( m\right) $ of the aforementioned problem. In addition, we describe
the limit of some normalized associated eigenfunctions when $k=1$.
