  We show that for a strongly convergent sequence of geometrically finite
Kleinian groups with geometrically finite limit, the Cannon-Thurston maps of
limit sets converge uniformly. If however the algebraic and geometric limits
differ, as in the well known examples due to Kerckhoff and Thurston, then
provided the geometric limit is geometrically finite, the maps of limit sets
converge pointwise but not uniformly.
