  This paper is concerned with the ergodic problem for superquadratic viscous
Hamilton-Jacobi equations with exponent m \textgreater{} 2. We prove that the
generalized principal eigenvalue of the equation converges to a constant as m
$\rightarrow$ $\infty$, and that the limit coincides with the generalized
principal eigenvalue of an ergodic problem with gradient constraint. We also
investigate some qualitative properties of the generalized principal eigenvalue
with respect to a perturbation of the potential function. It turns out that
different situations take place according to m = 2, 2 \textless{} m \textless{}
$\infty$, and the limiting case m = $\infty$.
