In this paper, we study the rarefaction wave case of the regularized Riemann problem proposed by Chu, Hong and Lee in SIMA MMS, 2020, for compressible Euler equations with a small parameter $$\nu $$ . The solutions $$\rho _\nu $$ and $$v_\nu $$ of such problems stand for the density and velocity of gas flow near vacuum, respectively. We show that as $$\nu $$ approaches 0, the solutions $$\rho _\nu $$ and $$v_\nu $$ converge to the solutions $$\rho $$ and v, respectively, of pressureless compressible Euler equations in $$L^1$$ sense. In addition, the $$L^1$$ convergence rates of these physical quantities in terms of $$\nu $$ are also investigated. The $$L^1$$ convergences and convergence rates are proved by two facts. One is to invent an a priori estimate coupled with the iteration method to the high-order derivatives of Riemann invariants so that we obtain the uniform boundedness of $$\partial _{x}^{i} \rho _{\nu }$$ ( $$i=0,1,2$$ ) and $$\partial _{x}^{j} v_{\nu }$$ ( $$j=0,1,2,3$$ ) on the requisite regions. The other is about convexity of characteristic curves, which is used to estimate the distances among characteristic curves in terms of $$\nu $$ . These theoretic results are also supported by numerical simulations.