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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

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The limit of $f(x^3)$ and the limit of $f(x)$ both exist therefore the limit of the difference of the two functions exist as well – user43418 Oct 4 '12 at 20:03

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 $$f(x)=\operatorname{sign}(x)=\left\{ \begin{array}{cc} 1 & x>0\\ 0 & x=0\\ -1 & x<1\end{array}\right.$$

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Hint: Can you show: If $g$ is continuous around $a$ and $g(a)=b$ and $\lim_{x\to b} f(x)$ exists, then $\lim_{x\to a} f(g(x))=\lim_{x\to b} f(x)$.

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Ok I proofed that. Could you help me now with the second part of the question – user43418 Oct 4 '12 at 22:23

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