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Question Number 97199 by bobhans last updated on 07/Jun/20

$$\underset{x\to 0}{\lim }\frac{\sin \mathrm{x}-\tan \mathrm{x}}{(\sqrt[3]{1+{\mathrm{x}}^2}-1)(\sqrt{1+\sin \mathrm{x}}-1)}=?$$

Answered by john santu last updated on 07/Jun/20

$$\underset{x\to 0}{\lim }\frac{\tan \left(x\right)(\cos x-1)}{(\sqrt[3]{1+x^2}-1)(\sqrt{1+\sin x}-1)}=$$$$\underset{x\to 0}{\lim }\frac{\tan \left(x\right)(-2\sin {}^2\left(\frac{x}{2}\right))}{(1+\frac{x^2}{3}-1)(1+\frac{\sin \left(x\right)}{2}-1)}=$$$$\underset{x\to 0}{\lim }\frac{-2\times \frac{1}{4}{x}^3}{\frac{1}{6}{x}^3}=-\frac{1}{2}\times \frac{6}{1}=-3$$

Commented by bobhans last updated on 07/Jun/20

$$\mathrm{yesss}....\mathrm{sir}$$

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