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Question Number 84377 by john santu last updated on 12/Mar/20

$$\underset{x\to 0}{\lim }\frac{\sqrt{1+\tan \mathrm{x}}-\sqrt{1+\sin \mathrm{x}}}{{\mathrm{x}}^2\sin \mathrm{x}}$$

Answered by john santu last updated on 12/Mar/20

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

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