  We apply G-convergence theory to study the asymptotic of the eigenvalue
problems of positive definite bounded self-adjoint $h$-dependent operators as
$h\to\infty$. Two operators are considered; a second order elliptic operator
and a general linear operator. Using the definition of G-convergence of
elliptic operator, we review convergence results of the elliptic eigenvalue
problem as $h\to\infty$. Also employing the general definition of G-convergence
of positive definite self-adjoint operator together with $\Gamma$-convergence
of the associated quadratic form, we characterize the G-limit as $h\to\infty$
of the general operator with some classes of perturbations. As a consequence,
we also prove the convergence of the corresponding spectrum.
