Summary: We consider the continuum limit of the relativistic Toda lattice. In particular, we propose a method in order to ’integrate’ this system of nonlinear partial differential equations for some particular initial data and boundary conditions, before possible shocks. First, we recall the relation between the finite relativistic Toda lattice and the theory of discrete Laurent orthogonal polynomials. Our analysis is then based on some results for the asymptotic theory of discrete Laurent orthogonal polynomials with varying recurrence coefficients and the connection with a constrained and weighted extremal problem for logarithmic potentials.