  Grosjean proved that the $(1/p)$-th power of the first eigenvalue of the
$p$-Laplacian on a closed Riemannian manifold converges to the twice of the
inverse of the diameter of the space, as $p \to \infty$. Before this, a
corresponding result for the Dirichlet first eigenvalues was also obtained by
Juutinen, Lindqvist and Manfredi. We extend those results for certain $k$-th
min-max value related to the $p$-energy, where the corresponding limits are
packing radii introduced by Grove-Markvorsen or its variant. Furthermore, we
remark that our result holds for more singular setting.
