  A powerful technique to calculate gravitational radiation from binary systems
involves a perturbative expansion: if the masses of the two bodies are very
different, the "small" body is treated as a point particle of mass $m_p$ moving
in the gravitational field generated by the large mass $M$, and one keeps only
linear terms in the small mass ratio $m_p/M$. This technique usually yields
finite answers, which are often in good agreement with fully nonlinear
numerical relativity results, even when extrapolated to nearly comparable mass
ratios. Here we study two situations in which the point-particle approximation
yields a divergent result: the instantaneous flux emitted by a small body as it
orbits the light ring of a black hole, and the total energy absorbed by the
horizon when a small body plunges into a black hole. By integrating the
Teukolsky (or Zerilli/Regge-Wheeler) equations in the frequency and time
domains we show that both of these quantities diverge. We find that these
divergences are an artifact of the point-particle idealization, and are able to
interpret and regularize this behavior by introducing a finite size for the
point particle. These divergences do not play a role in black-hole imaging,
e.g. by the Event Horizon Telescope.
