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

\lim_{x \to 0} \frac{\sin x \cos x}{\sqrt{π + 2 \sin x} - \sqrt{π}} ?

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

\lim_{x \to 0} \frac{x c o s x}{π + 2 s i n x - π} (\sqrt{π + 2 s i n x} + \sqrt{π})

= \frac{x}{2 s i n x} (\sqrt{π} + \sqrt{π}) (s i n x \to x a n d x \to 0)

= \frac{2 \sqrt{π}}{2} = \sqrt{π}

Answered by liberty last updated on 09/Nov/20

\lim_{x \to 0} \frac{\sin x \cos x}{\sqrt{π} (\sqrt{1 + \frac{2 \sin x}{π}} - 1)} =

\lim_{x \to 0} \frac{\cos x}{\sqrt{π}} . \frac{\sin x}{(1 + \frac{\sin x}{π}) - 1} =

\frac{1}{\sqrt{π}} \times \lim_{x \to 0} \frac{π \sin x}{\sin x} = \frac{π}{\sqrt{π}} = \sqrt{π} .

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