I need to prove that $\lim_{x \to 0}f(x^3)=\lim_{x \to 0}f(x)$. Then give an example of a function f for which $\lim_{x \to 0}f(x^2)$ exists but $\lim_{x \to 0}f(x)$ does not exist
Thank you in advance
I need to prove that $\lim_{x \to 0}f(x^3)=\lim_{x \to 0}f(x)$. Then give an example of a function f for which $\lim_{x \to 0}f(x^2)$ exists but $\lim_{x \to 0}f(x)$ does not exist
Thank you in advance
Hint: For the first part, do you know that $x\mapsto x^3$ is invertible on $\mathbb{R}$?
For the second part consider the function $f(x)=\operatorname{sign}(x)=\left\{ \begin{array}{cc} 1 & x>0\\ 0 & x=0\\ -1 & x<1\end{array}\right.$
Hint: Can you show: If $g$ is continuous around $a$ and $g(a)=b$ and $\lim_{x\to b} f(x)$ exists, then $\lim_{x\to a} f(g(x))=\lim_{x\to b} f(x)$.