In presence of linear diffusion and non-positive dispersion, we prove well-posedness of the nonlinear conservation equation $u_t + f(u)_x = \epsilon u_{xx} - \delta (u_{xx}^2)_x$. Then, as the right-hand perturbations vanish, we prove convergence of the previous solutions to the entropy weak solution of the hyperbolic conservation law $u_t+f(u)_x =0$.