  We consider perturbative solutions in Einstein gravity with higher-derivative
extensions and address some subtle issues of taking extremal limit. As a
concrete new result, we construct the perturbative rotating black hole in five
dimensions with equal angular momenta $J$ and general mass $M$ in
Einstein-Gauss-Bonnet gravity, up to and including the linear order of the
standard Gauss-Bonnet coupling constant $\alpha$. We obtain the near horizon
structure of the near extremal solution, with the blackening factor of the
order $\alpha$. In the extremal limit, the mass-angular momentum relation
reduces to $M=\frac32 \pi^{\frac13} J^{\frac23} + \pi \alpha$. The positive
sign of the $\alpha$-correction implies that the centrifugal repulsion
associated with rotations becomes weaker than the gravitational attraction
under the unitary requirement for the Gauss-Bonnet term.
