  Consider a nonlocal conservation where the flux function depends on the
convolution of the solution with a given kernel. In the singular local limit
obtained by letting the convolution kernel converge to the Dirac delta one
formally recovers a conservation law. However, recent counter-examples show
that in general the solutions of the nonlocal equations do not converge to a
solution of the conservation law. In this work we focus on nonlocal
conservation laws modeling vehicular traffic: in this case, the convolution
kernel is anisotropic. We show that, under fairly general assumptions on the
(anisotropic) convolution kernel, the nonlocal-to-local limit can be rigorously
justified provided the initial datum satisfies a one-sided Lipschitz condition
and is bounded away from $0$. We also exhibit a counter-example showing that,
if the initial datum attains the value $0$, then there are severe obstructions
to a convergence proof.
