  We study the behaviour of rational curves tangent to a hypersurface under
degenerations of the hypersurface. Working within the framework of logarithmic
Gromov-Witten theory, we extend the degeneration formula to the logarithmically
singular setting, producing a virtual class on the space of maps to the
degenerate fibre. We then employ logarithmic deformation theory to express this
class as an obstruction bundle integral over the moduli space of ordinary
stable maps. This produces new refinements of the logarithmic Gromov-Witten
invariants, encoding the degeneration behaviour of tangent curves. In the
example of a smooth plane cubic degenerating to the toric boundary we employ
localisation and tropical techniques to compute these refinements. Finally, we
leverage these calculations to describe how embedded curves tangent to a smooth
cubic degenerate as the cubic does; the results obtained are of a classical
nature, but the proofs make essential use of logarithmic Gromov-Witten theory.
