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Evaluate the given limit

Compute

    \[ \lim_{x \to 0} x \sin \frac{1}{x}. \]


Since -1 \leq \sin x \leq 1 for all x we have

    \[ -x \leq x \sin \frac{1}{x} \leq x \]

for all x. Since \lim_{x \to 0} x = 0, we have

    \[ \lim_{x \to 0} x \sin \frac{1}{x} = 0 \]

by the squeeze theorem (Theorem 3.3 in Apostol).

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