  We calculate semileptonic form factors for the decays $B_c \to \eta_c \, l
\nu$ and $B_c \to J/\psi \, l \nu$ over the entire $q^2$ range, using a highly
improved lattice quark action for charm at several lattice spacings down to
$a=0.045$ fm. We have two ways of treating the $b$ quark: either with an
$O(\alpha_s)$ improved NRQCD formalism or by extrapolating a heavy mass $m_h$
to $m_b$ in the relativistic formalism. Comparison of the two approaches
provides an important cross-check of methodologies in lattice QCD.
Nonperturbative renormalisation of the currents in the relativistic theory also
allows us then to fix NRQCD-charm normalisation for $b$ to $c$ decays such as
$B \to D$ and $B \to D^*$.
