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Question

Prove that $lim x \to 0 x sin \frac{1}{x} = 0$ .

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Solution

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For $x \in R - {0}$ ,

$- 1 \leq sin \frac{1}{x} \leq 1$

or, $- x \leq x sin \frac{1}{x} \leq x$

This gives,

$lim x \to 0 x sin \frac{1}{x} = 0$ .

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