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Question Number 98544 by abony1303 last updated on 14/Jun/20

Commented by abony1303 last updated on 14/Jun/20

$$\mathrm{Pls}\mathrm{help}$$

Answered by mathmax by abdo last updated on 14/Jun/20

$${\lim }_{\mathrm{x}\to 0}\frac{\mathrm{f}\left(\mathrm{x}\right)-\mathrm{f}\left(0\right)}{\mathrm{x}-0}={\lim }_{\mathrm{x}\to 0}\frac{{\mathrm{x}}^2\sin \left(\frac{1}{\mathrm{x}}\right)+0}{\mathrm{x}}={\lim }_{\mathrm{x}\to 0}\mathrm{xsin}\left(\frac{1}{\mathrm{x}}\right)=0$$$$\mathrm{because}\mid \mathrm{xsin}\left(\frac{1}{\mathrm{x}}\right)\mid ⩽\mid \mathrm{x}\mid \Rightarrow \mathrm{f}\mathrm{is}\mathrm{derivable}\mathrm{at}0\mathrm{and}{\mathrm{f}}^{{}^{\prime }}\left(0\right)=0$$

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