If $\gamma$ is a parametrized curve, its first derivative $\dot{\gamma}(t)$ is called the tangent vector of $\gamma$ at the point $\gamma(t)$.
So for a given paramaterized curve(assuming curve passing through origin), if I wanted to find tangent vector at origin,then I will find $t$ such that $\gamma(t)$ is origin and hence compute $\dot{\gamma}(t)$ as the tangent vector at origin.
My question is what if there are two points $t_1$ and $t_2$ such that $\gamma(t_1)=\gamma(t_2)=0(origin)$ and $\dot{\gamma}(t_1) \neq \dot{\gamma}(t_2)$ what will the tangent vector at origin in this case?
Well, in a way they are both tangent vectors at the origin, since the curve pass two times through it. That is why, to avoid this kind of ambiguity, one usually ask the curve $\gamma$ to be injective (i.e. no crossing allowed).