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Question Number 99676 by bemath last updated on 22/Jun/20

$$\underset{x\to 0}{\lim }\frac{\sin \left(\sin \mathrm{x}\right)-\mathrm{x}}{\mathrm{x}(\cos \left(\sin \mathrm{x}\right)-1)}??$$

Answered by bobhans last updated on 23/Jun/20

$$\underset{x\to 0}{\lim }\frac{\sin \mathrm{x}-\frac{\sin {}^3\mathrm{x}}{6}-\mathrm{x}}{\mathrm{x}(1-\frac{\sin {}^2\mathrm{x}}{2}-1)}=\underset{x\to 0}{\lim }\frac{\mathrm{x}-\frac{{\mathrm{x}}^3}{6}-\frac{\sin {}^3}{6}-\mathrm{x}}{\mathrm{x}(-\frac{\sin {}^2\mathrm{x}}{2})}$$$$=\underset{x\to 0}{\lim }\frac{-\frac{{\mathrm{x}}^3}{3}}{-\frac{{\mathrm{x}}^3}{2}}=\frac{1}{3}\times \frac{2}{1}=\frac{2}{3}◼$$

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