Does a curve $C \subset X$ that contains a singularity of $X$ pick up extra multiplicities?

Question

Suppose $X$ is a variety over a algebraically closed field, and $C \subset X$ is a curve, that contains a singularity of $x$ of $X$. If $D$ is a Cartierdivisor, whoose support $|D|$ contains $x$, is it true that $D|_C$ vanishes at $x$ with multiplicity $\geq 2$?

Answer 1

No. For example, let $X \subset \mathbb{P}^3$ be a quadratic cone, let $C \subset X$ be its ruling, and let $D = H \cap X$, where $H \subset \mathbb{P}^3$ is a hyperplane passing through the vertex of the cone and transverse to $C$. Then $$ D \vert_C = (H \vert_X) \vert_C = H\vert_C $$ has multiplicity 1.