In this paper we prove that if Toeplitz operators $T^{alpha}_u$ with symbols in RW satisfy ${parallel}uk^{alpha}_z{parallel}_{s,{alpha}{ ightarrow}0$ as $z{ ightarrow}{partial}mathbb{D}$ then $T^{alpha}_u$ is compact and also prove that if $T^{alpha}_u$ is compact then the Berezin transform of $T^{alpha}_u$ equals to zero on ${partial}mathbb{D}$.