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Question Number 126672 by shaker last updated on 23/Dec/20

Answered by liberty last updated on 23/Dec/20

$$-1⩽\sin \left(\frac{3}{x}\right)⩽1;-(x+4)⩽(x+4)\sin \left(\frac{3}{x}\right)⩽x+4$$$$\underset{x\to 0}{\lim }-(x+4)⩽\underset{x\to 0}{\lim }(x+4)\sin \left(\frac{3}{x}\right)⩽\underset{x\to 0}{\lim }(x+4)$$$$\underset{x\to 0}{\lim }-(x+4)=-4$$$$\underset{x\to 0}{\lim }(x+4)=4$$$$-4⩽\underset{x\to 0}{\lim }(x+4)\sin \left(\frac{3}{x}\right)⩽4$$$$$$

Answered by mathmax by abdo last updated on 24/Dec/20

$$(\mathrm{x}+4)\sin \left(\frac{3}{\mathrm{x}}\right)=\mathrm{xsin}\left(\frac{3}{\mathrm{x}}\right)+4\sin \left(\frac{3}{\mathrm{x}}\right){\lim }_{\mathrm{x}\to 0}\mathrm{xsin}\left(\frac{3}{\mathrm{x}}\right)=0\mathrm{but}$$$$\sin \left(\frac{3}{\mathrm{x}}\right)\mathrm{hsvent}\mathrm{any}\mathrm{limit}\mathrm{at}0!$$

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