  We study a family of initial boundary value problems associated to mixed
hyperbolic-parabolic systems:
  v^{\epsilon} _t + A (v^{\epsilon}, \epsilon v^{\epsilon}_x ) v^{\epsilon}_x =
\epsilon B (v^{\epsilon} ) v^{\epsilon}_{xx}
  The conservative case is, in particular, included in the previous
formulation.
  We suppose that the solutions $v^{\epsilon}$ to these problems converge to a
unique limit. Also, it is assumed smallness of the total variation and other
technical hypotheses and it is provided a complete characterization of the
limit.
  The most interesting points are the following two.
  First, the boundary characteristic case is considered, i.e. one eigenvalue of
$A$ can be $0$.
  Second, we take into account the possibility that $B$ is not invertible. To
deal with this case, we take as hypotheses conditions that were introduced by
Kawashima and Shizuta relying on physically meaningful examples. We also
introduce a new condition of block linear degeneracy. We prove that, if it is
not satisfied, then pathological behaviours may occur.
