We study the behavior of the Aw-Rascle-Zhang model when the relaxation parameter converges to zero. In a Lagrangian setting, we use the Wave-Front-Tracking method with splitting technique to construct a sequence of approximate solutions. We prove that this sequence converges to a weak entropy solution of the relaxed system associated to a given initial datum with bounded variation. Besides, we also provide an estimate on the decay of positive waves. We finally prove that the solutions of the Aw-Rascle-Zhang system with relaxation converge to a weak solution of the corresponding scalar conservation law when the relaxation parameter goes to zero.