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Question Number 116588 by bemath last updated on 05/Oct/20

$$\underset{x\to 0}{\lim }\frac{4\mathrm{x}}{2\mathrm{cosec}\mathrm{x}(1-\sqrt{\cos \mathrm{x}})}=?$$

Answered by bobhans last updated on 05/Oct/20

$$\underset{x\to 0}{\lim }\frac{4\mathrm{x}.\sin \mathrm{x}}{2(1-\sqrt{1-2\sin {}^2\left(\frac{\mathrm{x}}{2}\right)})}=$$$$\underset{x\to 0}{\lim }\frac{4\mathrm{x}.\sin \mathrm{x}}{2(1-(1-\frac{2\left(\frac{{\mathrm{x}}^2}{4}\right)}{2}))}=$$$$\underset{x\to 0}{\lim }\left[\frac{{2\mathrm{x}}^2}{\left(\frac{{\mathrm{x}}^2}{4}\right)}\right]=8$$

Answered by Dwaipayan Shikari last updated on 05/Oct/20

$$\underset{x\to 0}{\lim }\frac{{4\mathrm{x}}^2}{2(1-\sqrt{\mathrm{cosx}})}=\frac{{2\mathrm{x}}^2}{1-\mathrm{cosx}}(1+\sqrt{\mathrm{cosx}})$$$$=2.\frac{{2\mathrm{x}}^2}{{2\sin }^2\frac{\mathrm{x}}{2}}=8$$

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