  In complex systems, it is quite common to resort to approximations when
optimizing system performance. These approximations typically involve selecting
a particular system parameter and then studying the performance of the system
as this parameter grows without bound. In such an asymptotic regime, we prove
that if the approximation to the objective function is accurate up to
$\mathcal{O}(1)$, then under some regularity conditions, the prescriptions that
are derived from this approximation are $o(1)$-optimal, i.e., their optimality
gap is asymptotically zero. A consequence of this result is that the well-known
square-root staffing rules for capacity sizing in $M/M/s$ and $M/M/s+M$ queues
to minimize the sum of linear expected steady-state customer waiting costs and
linear capacity costs are $o(1)$-optimal. We also discuss extensions of this
result for the case of non-linear customer waiting costs in these systems.
