  We study QCD in 1+1 dimensions in the large $N_c$ limit using light-front
Hamiltonian perturbation theory in the $1/N_c$ expansion. We use this formalism
to exactly compute hadronic transition matrix elements for arbitrary currents
at leading order in $1/N_c$. We compute the semileptonic differential decay
rate of a heavy meson, $d\Gamma/dx$, and its moments, $M_N$, using the hadronic
matrix elements obtained previously. We put some emphasis in trying to
understand parity invariance. We also study with special care the kinematic
region where the operator product expansion ($1/N \sim 1-x \sim 1$) or
non-local effective field theories ($1/N \sim 1-x \sim \Lambda_{QCD}/m_Q$) can
be applied. We then compare with the results obtained using an effective field
theory approach based on perturbative factorization, with the focus to better
understand quark-hadron duality. At the end of the day, using effective field
theories, we have been able to obtain expressions for the moments with relative
accuracy of $O(\Lambda_{QCD}^2/m_Q^2)$ in the kinematic region where the
operator product expansion can be applied, and with relative accuracy of
$O(\Lambda_{QCD}/m_Q)$ in the kinematic region where non-local effective field
theories can be applied. These expressions agree, within this precision, with
those obtained from the hadronic result using the layer-function approximation
plus Euler-McLaurin expansion. Very good numerical agreement for the moments is
obtained between the exact result and the result using effective field
theories.
