Let: $$\displaystyle{ A = \left\{ (x,y) \in \mathbb{R}^2: x\left( x^2-4y^2 \right)\left( x-y^3 \right) = 0 \right\}}$$Find all vectors tangent to the set $A$ in the point $(0,0)$
I know that vector $v$ is tangent to the set $A$ in the point $a$ when exist $x_n \in A\backslash \left\{a\right\}$ such that $x_n \to a$ and $\lim_{n \to +\infty} \frac{x_n-a}{||x_n-a||}=\frac{v}{||v||}$.
However I don't know how to use this knowledge in this task because so far in the tasks I determined $y$ depending on $x$, but in this task it is too difficult and there is definitely a better way.
hint
$$A=\{(x,y)\in \Bbb R^2 \;\; x=0 \vee y=\frac x2 \vee y=-\frac x2 \vee y=x^\frac 13\}$$
thus there are three tangents.
$$x=0$$
$$y=\frac 12 x$$ $$y=-\frac 12 x$$