In [9], K-T. Kim gave a nice characterization of a domain in $\mathbb C^n$ satisfying the Condition $(L)$ at a boundary point. The key point in his proof is the convergence of the certain scaling sequence. Unfortunately, this assertion is not true. The purpose of this article is twofold. The first is to show a counterexample to the convergence of Kim’s scaling sequence. The second is to give a correct proof of Kim’s characterization theorem.