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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 highlyimproved 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 approachesprovides an important cross-check of methodologies in lattice QCD.Nonperturbative renormalisation of the currents in the relativistic theory alsoallows us then to fix NRQCD-charm normalisation for $b$ to $c$ decays such as$B \to D$ and $B \to D^*$.

$B_c$ decays from highly improved staggered quarks and NRQCD