Tangent space of an immersed submanifold

Question

If $M$ is a smooth manifold and $S$ is an embedded submanifold, then for any point $p \in S$, the tangent space $T_p(S)$ is characterized as $T_p(S) = \{v \in T_p(M) \colon v(f) = 0, f \in C^{\infty}(M)\}$, where $f$ restricted to $S$ is just the $0$ fuction and $T_p{M}$ is the tangent space to $M$ at $p$. Now, this characterization of the tangent space isn't true for immersed subamnifolds of $M$ and I'm looking for a counter example. I was thinking about taking $M = \mathbb{R}^2$ and $S$ to be the Figure-8 space and $p$ to be the point $(0,0)$, but I'm not sure how to compute the tangent space at that point.