  We show convergence of small eigenvalues for geometrically finite hyperbolic
$n$-manifolds under strong limits. For a class of convergent convex sets in a
strongly convergent sequence of Kleinian groups, we use the spectral gap of the
limit manifold and the exponentially mixing property of the geodesic flow along
the strongly convergent sequence to find asymptotically uniform counting
formulas for the number of orthogeodesics between the convex sets. In
particular, this provides asymptotically uniform counting formulas (with
respect to length) for orthogeodesics between converging Margulis tubes,
geodesic loops based at converging basepoints, and primitive closed geodesics.
