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We consider perturbative solutions in Einstein gravity with higher-derivativeextensions and address some subtle issues of taking extremal limit. As aconcrete new result, we construct the perturbative rotating black hole in fivedimensions with equal angular momenta $J$ and general mass $M$ inEinstein-Gauss-Bonnet gravity, up to and including the linear order of thestandard Gauss-Bonnet coupling constant $\alpha$. We obtain the near horizonstructure of the near extremal solution, with the blackening factor of theorder $\alpha$. In the extremal limit, the mass-angular momentum relationreduces to $M=\frac32 \pi^{\frac13} J^{\frac23} + \pi \alpha$. The positivesign of the $\alpha$-correction implies that the centrifugal repulsionassociated with rotations becomes weaker than the gravitational attractionunder the unitary requirement for the Gauss-Bonnet term.

$D=5$ Rotating Black Holes in Einstein-Gauss-Bonnet Gravity: Mass and Angular Momentum in Extremality