In the article "Stability of Rarefaction Waves to the 1D Compressible Navier–Stokes Equations with Density-Dependent Viscosity" (DOI: 10.1080/03605302.2010.516785), in page 619, it get the limit $$\lim_{\rho\to0}\frac{|(\rho^{\theta-\frac{1}{2}}-\bar{\rho}^{\theta-\frac{1}{2}})|^2\mathbb{1}_{\{0\leq \rho\leq 2\rho_+\}}}{\rho\Psi(\rho,\bar\rho)}=0$$
$\rho$ is the solution of the equations $$ \left\{ \begin{array}{l} \rho_t+(\rho u)_x=0\\ (\rho u)_t+(\rho u^2+p(\rho))_x=(\mu_\varepsilon(\rho)u_x)_x \end{array} \right. $$
where $p(\rho)=\rho^\gamma,\gamma>1,\mu_\varepsilon(\rho)=\rho^\alpha+\varepsilon\rho^\theta,\alpha>0,\varepsilon>0,\theta\in(0,\frac{1}{2})$
$\bar\rho$ is the solution of the equations $$ \left\{ \begin{array}{l} \bar\rho_t+(\bar\rho \bar u)_x=0\\ (\bar\rho \bar u)_t+(\bar\rho \bar u^2+p(\bar\rho))_x=0 \end{array} \right. $$
$\rho\Psi(\rho,\bar\rho)=(\rho^\gamma-\bar\rho^\gamma-\gamma\bar\rho^{\gamma-1}(\rho-\bar\rho))/(\gamma-1)$
In the article, $\bar\rho$ satisfies $0<\rho_-\leq\bar\rho\leq\rho_+$
So why the limit can hold?