  We deal with heteroclinic planar fronts for parameter-dependent
reaction-diffusion equations with bistable reaction and saturating diffusive
term like $$ u_t=\epsilon \, \textrm{div}\, \left(\frac{\nabla u}{\sqrt{1+\vert
\nabla u \vert^2}}\right) + f(u), \quad u=u(x, t), \; x \in \textbf{R}^n, \, t
\in \textbf{R}, $$ analyzing in particular their behavior for $\epsilon \to 0$.
First, we construct monotone and non-monotone planar traveling waves, using a
change of variables allowing to analyze a two-point problem for a suitable
first-order reduction; then, we investigate their asymptotic behavior for
$\epsilon \to 0$, showing in particular that the convergence of the critical
fronts to a suitable step function may occur passing through discontinuous
solutions.
