  We introduce a "limiting Frobenius structure" attached to any degeneration of
projective varieties over a finite field of characteristic p which satisfies a
p-adic lifting assumption. Our limiting Frobenius structure is shown to be
effectively computable in an appropriate sense for a degeneration of projective
hypersurfaces. We conjecture that the limiting Frobenius structure relates to
the rigid cohomology of a semistable limit of the degeneration through an
analogue of the Clemens-Schmidt exact sequence. Our construction is
illustrated, and conjecture supported, by a selection of explicit examples.
