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Question Number 100036 by bobhans last updated on 24/Jun/20

$$\mathrm{f}\left(\mathrm{x}\right)=\frac{\sqrt{1+\sin 2\mathrm{x}}-\sqrt{1-2\sin \mathrm{x}}}{\mathrm{x}}$$$$\mathrm{g}\left(\mathrm{x}\right)=2\mathrm{x}+\sqrt{2\mathrm{x}}.\mathrm{find}\underset{x\to 0}{\lim }\mathrm{g}\left(\mathrm{f}\left(\mathrm{x}\right)\right)$$

Answered by john santu last updated on 24/Jun/20

$$\underset{x\to 0}{\lim }\mathrm{g}\left(\mathrm{f}\left(\mathrm{x}\right)\right)=\mathrm{g}\left(\underset{x\to 0}{\lim }\mathrm{f}\left(\mathrm{x}\right)\right)$$$$=\mathrm{g}\left(\underset{x\to 0}{\lim }\frac{\sqrt{1+\sin 2\mathrm{x}}-\sqrt{1-2\sin \mathrm{x}}}{\mathrm{x}}\right)$$$$=\mathrm{g}\left(\underset{x\to 0}{\lim }\frac{(1+\frac{\sin 2\mathrm{x}}{2})-(1-\frac{2\sin \mathrm{x}}{2})}{\mathrm{x}}\right)$$$$=\mathrm{g}\left(\underset{x\to 0}{\lim }\frac{\frac{1}{2}\sin 2\mathrm{x}+\sin \mathrm{x}}{\mathrm{x}}\right)=\mathrm{g}\left(2\right)$$$$=2.2+\sqrt{2.2}=4+2=6◼$$

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