The goal of this article is to develop a new technique to obtain better asymptotic estimates for scalar conservation laws. General convex flux, f(n) (u) greater than or equal to 0, is considered with an assumption lim(u-->0) uf'(u)/f (u) = gamma > 1. We show that, under suitable conditions on the initial value, its solution converges to an N-wave in L-1 norm with the optimal convergence order of 0(1/t). The technique we use in this article is to enclose the solution with two rarefaction waves. We also show a uniform convergence order in the sense of graphs. A numerical example of this phenomenon is included. (C) 2003 Elsevier Science (USA). All rights reserved.