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Given any function $f: \mathbb{R^n} \to \mathbb{R^m}$ , if $$\lim_{x\rightarrow a}\|f(x)\| = 0$$ then does $$\lim_{x\rightarrow a}\frac{\|f(x)\|}{\|x-a\|} = 0 $$

as well? Is the converse true?

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    Take $f(x) = \sqrt{|x|}$, then $\lim_{x \to 0} |f(x)| = 0$, but $|\frac{f(x)}{x}| \to \infty$, as $x \to 0$.2012-05-25

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