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Question Number 121527 by benjo_mathlover last updated on 09/Nov/20

$$\underset{x\to 0}{\lim }\frac{\sin \mathrm{x}\cos \mathrm{x}}{\sqrt{\pi +2\sin \mathrm{x}}-\sqrt{\pi }}?$$

Answered by Dwaipayan Shikari last updated on 09/Nov/20

$$\underset{x\to 0}{\lim }\frac{xcosx}{\pi +2sinx-\pi }(\sqrt{\pi +2sinx}+\sqrt{\pi })$$$$=\frac{x}{2sinx}(\sqrt{\pi }+\sqrt{\pi })(sinx\to xandx\to 0)$$$$=\frac{2\sqrt{\pi }}{2}=\sqrt{\pi }$$

Answered by liberty last updated on 09/Nov/20

$$\underset{x\to 0}{\lim }\frac{\sin \mathrm{x}\cos \mathrm{x}}{\sqrt{\pi }(\sqrt{1+\frac{2\sin \mathrm{x}}{\pi }}-1)}=$$$$\underset{x\to 0}{\lim }\frac{\cos \mathrm{x}}{\sqrt{\pi }}.\frac{\sin \mathrm{x}}{(1+\frac{\sin \mathrm{x}}{\pi })-1}=$$$$\frac{1}{\sqrt{\pi }}\times \underset{x\to 0}{\lim }\frac{\pi \sin \mathrm{x}}{\sin \mathrm{x}}=\frac{\pi }{\sqrt{\pi }}=\sqrt{\pi }.$$

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