For any convergent sequence of Riemannian spaces, it is
possible to extract a subsequence for which their corresponding
tangent bundles converge as well. These limits sometimes coincide
with preexisting notions of tangency, but not always. In the process
of understanding the structure of the limiting space, a couple of
natural elementary constructions are introduced at the level of
the individual Riemannian spaces. Lastly, a weak notion of parallelism is
discussed for the limits.