  A comparative study of the full $\alpha_s^4$ perturbative QCD calculation of
the hadronic production of the $B_c$ and $B_c^*$ mesons and the fragmentation
approximation is presented. We examined the various subprocesses in detail and
the distribution $C(z)$, where $z$ is twice the fraction of the $B_c$ or
$B_c^*$ meson energy in the center of mass of the subprocess, which proved
insightful and revealed the importance of certain non-fragmentation
contributions. We concluded that the condition for the applicability of the
fragmentation approximation in hadronic collisions is that the transverse
momentum $P_T$ of the produced $B_c$ and $B_c^*$ be much larger than the mass
of the $B_c$ meson; i.e., $P_T\gg M_{B_c}$. Numerical results for the cross
sections at the Fermilab Tevatron are also presented using updated parton
distribution functions with various kinematic cuts.
