  In this paper, we study the critical Sobolev embeddings
$W^{1,p(x)}(\Omega)\subset L^{p^*(x)}(\Omega)$ for variable exponent Sobolev
spaces from the point of view of the $\Gamma$-convergence. More precisely we
determine the $\Gamma$-limit of subcritical approximation of the best constant
associated with this embedding. As an application we provide a sufficient
condition for the existence of extremals for the best constant.
