Let $X$ be a variety over $k$, $x$ be a closed point on $X$ with residue field $k$, and $v: Spec(k[\epsilon]) \to X$ be a tangent vector of $X$ at $x$, where $k[\epsilon]$ is the dual number algebra.

Then how to find an affine smooth curve $C$ with a point $p$ on it, and a morphism $ \gamma :C\to X$, a tangent vector $u: Spec(k[\epsilon])\to C$, such that

- $u(0)=p$, $\gamma(p)=x$
- $v$ factors through u.

Thank you very much!