How to prove that $\lim_{x\to 0}f(x)=L$ is equivalent to $\lim_{x\to 0}f(x^3)=L$ rigorously I've tried to prove this by epsilon-delta, but it didn't go well...

$$\forall \epsilon>0\,\exists \delta>0\,\forall x:\,|x|\le \delta\Rightarrow |f(x)-L|\le\epsilon.$$
For a given $\epsilon$ take $\delta_1=\min(\delta,1)$. After that $|x^3|<|x|$.
