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Question Number 119482 by liberty last updated on 24/Oct/20

$$\underset{x\to 0}{\lim }{x}^2\cos \left(\frac{1}{x}\right)?$$

Answered by Olaf last updated on 25/Oct/20

$$\mathrm{\forall }x\in {\mathbb{R}}^{\ast },-1⩽\cos \left(\frac{1}{x}\right)⩽+1$$$$\mathrm{\forall }x\in {\mathbb{R}}^{\ast },-x^2⩽x^2\cos \left(\frac{1}{x}\right)⩽+x^2$$$$\mathrm{And}\underset{x\to 0}{\lim }(\pm x^2)=0$$$$\Rightarrow \underset{x\to 0}{\lim }{x}^2\cos \left(\frac{1}{x}\right)=0$$

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