  Let $X$ be a smooth projective K3 surface over complex numbers and $C$ be an
ample curve on $X$. In this paper we will study the semistability of the
Lazarsfeld-Mukai bundle $E_{C, A}$ associated to a line bundle $A$ ion $C$ such
that $|A|$ is a pencil on $C$ and computes the Clifford index of $C$. We give a
necessary and sufficient condition for $E_{C, A}$ being semistable.
