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Prove that if one of $\displaystyle\lim_{x \rightarrow 0 }{f(x)}$ and $\displaystyle\lim_{x \rightarrow 0}{f(x^3)}$ exists, then the other one also exists. Can anyone guide me on this ? I have no idea on how to start .

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Just note that $x\longrightarrow 0$ if and only if $x^3\longrightarrow 0$. –  1015 Mar 17 '13 at 15:42

2 Answers 2

Hint: If $\lvert x^3 \rvert \le \delta^3$, with $\delta \le 1$, then $\lvert x \lvert \le \delta$ too. So you can go through the $\epsilon$ - $\delta$ dance for one and the other in parallel with minor modifications.

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Consider $\lim_{x \rightarrow 0}{f(x^3)}$ exists

Need to prove if $\lim_{x \rightarrow 0}{f(x^3)}$ exists $$\lim_{x \rightarrow 0}{f(x^3)}$$

Let $y = x^3$, then if $x\to0$, $y\to0$ $$\lim_{y \rightarrow 0}{f(y)}$$ But given $\lim_{x \rightarrow 0}{f(x)}$ exist so $\lim_{x \rightarrow 0}{f(x^3)}$ also exist

In similar way Consider $\lim_{x \rightarrow 0}{f(x)}$ exists Need to prove $\lim_{x \rightarrow 0}{f(x)}$ exists

Let $y = x^{1/3}$, then if $x\to0$, $y\to0$

so we have $$\lim_{y \rightarrow 0}{f(y^3)}$$ But $\lim_{x \rightarrow 0}{f(x)}$ exists so $\lim_{y \rightarrow 0}{f(y^3)}$ also exists

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i guess there are a lot of typos? –  Idonknow Mar 17 '13 at 16:18

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