Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function, and $b \in \mathbb{R}$. Prove that the $\lim_{x \rightarrow 0}f(x)=b$ is equivalent to the $\lim_{x \rightarrow 0}f(x^3)=b$.
I suppose that using the limit definition here is our best chance. Also, knowing that the same equivalence cannot hold for $f(x^2)$, I think there is a need for a symmetry argument.
Any help is welcome, thanks.
Recall (or show) that: If $g\colon \mathbb R\to\mathbb R$ is continuous at $a$, then $\lim_{x\to g(a)}f(x)=b$ implies $\lim_{x\to a}f(g(x))=b$. Apply this to $g(x)=x^3$ and to $g(x)=\sqrt[3]x$.