Why does tangent not exist at point $(0,0)$ of the curve $(x,y) = (t^3,t^5),~~ t\in \mathbb{R}$

Question

Given the curve $(x,y) = (t^3,t^5)$, why does such parametrisation does not give the tangent at $(0,0)$? and why would other parametrisation give a valid tangent? What would be such a valid paramatrization?

Answer 1

Your curve can also be parametrized as $(x,x^{\frac{5}{3}})$, so it is the graph of a $C^1$ function $f(x)\colon x\mapsto x^{\frac{5}{3}}$ from $\mathbb{R}$ to $\mathbb{R}$. The derivative $f'(x) = \frac{5}{3} x^{\frac{2}{3}}$ is $0$ at $0$, so the tangent line at $(0,0)$ is the $x$ axis.