  Previous work has shown that if an attractive 1/r^2 potential is regularized
at short distances by a spherical square-well potential, renormalization allows
multiple solutions for the depth of the square well. The depth can be chosen to
be a continuous function of the short-distance cutoff R, but it can also be a
log-periodic function of R with finite discontinuities, corresponding to a
renormalization group (RG) limit cycle. We consider the regularization with a
delta-shell potential. In this case, the coupling constant is uniquely
determined to be a log-periodic function of R with infinite discontinuities,
and an RG limit cycle is unavoidable. In general, a regularization with an RG
limit cycle is selected as the correct renormalization of the 1/r^2 potential
by the conditions that the cutoff radius R can be made arbitrarily small and
that physical observables are reproduced accurately at all energies much less
than hbar^2/mR^2.
