  The fact that one must evaluate the near-extremal and near-horizon limits of
Kerr space-time in a specific order, is shown to a lead to discontinuity in the
extremal limit, such that this limiting space-time differs nontrivially from
the precisely extremal space-time. This is established by first showing a
discontinuity in the extremal limit of the maximal analytic extension of the
Kerr geometry, given by Carter. Next, we examine the ISCO of the exactly
extremal Kerr geometry and show that on the event horizon of the extremal Kerr
black hole, it coincides with the principal null geodesic generator of the
horizon, having vanishing energy and angular momentum. We find that there is no
such ISCO in the near-extremal geometry, thus garnering additional support for
our primary contention. We relate this disparity between the two geometries to
the lack of a trapping horizon in the extremal situation.
