  The relations between most observables associated with a compact star, such
as the mass and radius of a neutron star or a quark star, typically depend
strongly on their unknown internal structure. The I-Love-Q relations (between
the moment of inertia, the tidal deformability and the quadrupole moment) are
however approximately insensitive to this structure. These relations become
exact for stationary black holes in General Relativity as shown by the no-hair
theorems. In this paper, we take the first steps toward studying how the
approximate I-Love-Q relations become exact in the limit as compact stars
become black holes. To do so, we consider a toy model, i.e. incompressible
stars with anisotropic pressure, which allows us to model an equilibrium
sequence of stars with their compactness approaching the black hole limit
arbitrarily closely. We extract the I-Love-Q trio by numerically constructing
such a sequence in the slow-rotation and small-tide approximations. We find
that the I-Love-Q relations approach the black hole limit in a nontrivial way,
with the quadrupole moment and the tidal deformability changing sign as the
compactness and the amount of anisotropy are increased. Generalizing Maclaurin
spheroids to anisotropic stars, we show that the multipole moments also change
sign in the Newtonian limit as the amount of anisotropy is increased. We also
prove analytically that the stellar moment of inertia reaches the black hole
limit as the compactness reaches the black hole value in the strongly
anisotropic limit. Modeling the black hole limit through a sequence of
anisotropic stars, however, fails when considering other theories of gravity.
We calculate the scalar dipole charge and the moment of inertia in a
parity-violating modified theory and find that these quantities do not tend to
their black hole counterparts as the anisotropic stellar sequence approaches
the black hole limit.
