Number of revolutions of a particle around a black hole: Is it infinite or finite?

We consider a particle falling into a rotating black hole. Such a particle makes an infinite number of revolutions $n$ from the viewpoint of a remote observer who uses the Boyer-Lindquist type of coordinates. We examine the behavior of $n$ when it is measured with respect to a local reference frame that also rotates due to dragging effect of spacetime. The crucial point consists here in the observation that for a nonextremal black hole, the leading contributions to $n$ from a particle itself and the reference frame have the same form being in fact universal, so that divergences mutually cancel. As a result, the relative number of revolutions turns out to be finite. For the extremal black hole this is not so, $n$ can be infinite. Different choices of the local reference frame are considered, the results turn out to be the same qualitatively. For illustration, we discuss two explicit examples - rotation in the flat spacetime and in the Kerr metric.