Given are $f: \mathbb{R} \to \mathbb{R}$ and a real number $b\in \mathbb{R}$. How can I prove that the statements $\lim_{x\to 0}f(x)=b$ and $\lim_{x\to 0}f(x^3)=b$ are equivalent?
So far I just have that I need to prove that if
1) $\lim_{x\rightarrow 0}f(x)=b$ then $\lim_{x\rightarrow 0}f(x^3)=b$
2) $\lim_{x\rightarrow 0}f(x^3)=b$ then $\lim_{x\rightarrow 0}f(x)=b$
And the definitions, so that if $\lim_{x\rightarrow 0}f(x)=b$ is true, then for $x\in\operatorname{dom}(f)$ and $|x-0|< \delta$ then $|f(x)-b|<\epsilon$
And if $\lim_{x\rightarrow 0}f(x^3)=b$ is true, then for each then for $x\in\operatorname{dom}(f)$ and $|x^3-0|< \delta $ then $|f(x^3)-b|<\epsilon$