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Question Number 116331 by bobhans last updated on 03/Oct/20

$$\underset{x\to 0}{\lim }\frac{3\mathrm{x}(\cos 7\mathrm{x}-\cos 3\mathrm{x})}{\sqrt{\tan 2\mathrm{x}+1}-\sqrt{\sin 2\mathrm{x}+1}}?$$

Answered by bemath last updated on 03/Oct/20

$$\underset{x\to 0}{\lim }(\sqrt{\tan 2\mathrm{x}+1}+\sqrt{\sin 2\mathrm{x}+1})\times \underset{x\to 0}{\lim }\frac{3\mathrm{x}(-2\sin 5\mathrm{x}.\sin 2\mathrm{x})}{\tan 2\mathrm{x}(1-\cos 2\mathrm{x})}$$$$=2\times \underset{x\to 0}{\lim }\frac{-6\mathrm{x}\sin 5\mathrm{x}\sin 2\mathrm{x}}{2\tan 2\mathrm{x}\sin {}^2\mathrm{x}}$$$$=\frac{-6.5.2}{2}=-30$$

Answered by Dwaipayan Shikari last updated on 03/Oct/20

$$\underset{x\to 0}{\lim 2}\frac{3\mathrm{x}(\cos 7\mathrm{x}-\cos 3\mathrm{x})}{\tan 2\mathrm{x}-\mathrm{sinx}}=\frac{6\mathrm{x}(-2\sin 5\mathrm{xsin}2\mathrm{x})}{\sin 2\mathrm{x}(1-\cos 2\mathrm{x})}=\frac{-12\mathrm{x}\left(5\mathrm{x}\right)}{{2\sin }^2\mathrm{x}}=-30$$

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