  We study the behavior of limits of tangents in topologically equivalent
spaces. In the context of families of generically reduced curves, we introduce
the $s$-invariant of a curve and we show that in a Whitney equisingular family
with the property that the $s$-invariant is constant along the parameter space,
the number of tangents of each curve of the family is constant. In the context
of families of isolated surface singularities, we show through examples that
Whitney equisingularity is not sufficient to ensure that the tangent cones of
the family are homeomorphic. We explain how the existence of exceptional
tangents is preserved by Whitney equisingularity but their number can change.
