  General relativistic quasiequilibrium states of black hole-neutron star
binaries are computed in the moving-puncture framework. We propose three
conditions for determining the quasiequilibrium states and compare the
numerical results with those obtained in the excision framework. We find that
the results obtained in the moving-puncture framework agree with those in the
excision framework and with those in the third post-Newtonian approximation for
the cases that (i) the mass ratio of the binary is close to unity irrespective
of the orbital separation, and (ii) the orbital separation is large enough
($m_0\Omega \alt 0.02$ where $m_0$ and $\Omega$ are the total mass and the
orbital angular velocity, respectively) irrespective of the mass ratio. For
$m_0 \Omega \agt 0.03$, both of the results in the moving-puncture and excision
frameworks deviate, more or less, from those in the third post-Newtonian
approximation. Thus the numerical results do not provide a quasicircular state,
rather they seem to have a nonnegligible eccentricity of order 0.01--0.1. We
show by numerical simulation that a method in the moving-puncture framework can
provide approximately quasicircular states in which the eccentricity is by a
factor of $\sim 2$ smaller than those in quasiequilibrium given by other
approaches.
