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Question Number 170182 by mathlove last updated on 18/May/22

$$\underset{x\to 0}{\lim }[x\cdot sin\frac{1}{x}]=?$$

Commented by mr W last updated on 18/May/22

$$has\left[\right]specialmeaning?oryoujustmean$$$$\underset{x\to 0}{\lim }(x\cdot sin\frac{1}{x})=?$$

Answered by mr W last updated on 18/May/22

$$-\mid x\mid ⩽x\sin \frac{1}{x}⩽\mid x\mid$$$$\underset{x\to 0}{\lim }(-\mid x\mid )⩽\underset{x\to 0}{\lim }\left(x\sin \frac{1}{x}\right)⩽\underset{x\to 0}{\lim }\mid x\mid$$$$0⩽\underset{x\to 0}{\lim }\left(x\sin \frac{1}{x}\right)⩽0$$$$\Rightarrow \underset{x\to 0}{\lim }\left(x\sin \frac{1}{x}\right)=0$$

Commented by mathlove last updated on 18/May/22

$$thanks$$

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