  Taking the extremal limit of a non-extremal Reissner-Nordstr\"om black hole
(by externally varying the mass or charge), the region between the inner and
outer event horizons experiences an interesting fate -- while this region is
absent in the extremal case, it does not disappear in the extremal limit but
rather approaches a patch of $AdS_2\times S^2$. In other words, the approach to
extremality is not continuous, as the non-extremal Reissner-Nordstr\"om
solution splits into two spacetimes at extremality: an extremal black hole and
a disconnected $AdS$ space. We suggest that the unusual nature of this limit
may help in understanding the entropy of extremal black holes.
