lim-x-pi-7-2sin-x-7-2sin-x-tan-x-

Question Number 124273 by bramlexs22 last updated on 02/Dec/20

\lim_{x \to π} \frac{\sqrt{7 + 2 \sin x} - \sqrt{7 - 2 \sin x}}{\tan x} = ?

Answered by Dwaipayan Shikari last updated on 02/Dec/20

\lim_{x \to π} \frac{\sqrt{7 + 2 (π - x)} - \sqrt{7 - 2 π + 2 x}}{π - x} = \sqrt{7} (\frac{\sqrt{1 + \frac{2}{7} (π - x)} - \sqrt{1 - \frac{2}{7} (π - x)}}{π - x})

= \sqrt{7} (\frac{1 + \frac{1}{7} (π - x) - 1 + \frac{1}{7} (π - x)}{π - x}) c o s x = - \sqrt{7} (\frac{2}{7}) = - \frac{2}{\sqrt{7}}

Answered by liberty last updated on 02/Dec/20

\lim_{x \to π} \frac{(7 + 2 \sin x) - (7 - 2 \sin x)}{(\sqrt{7 + 2 \sin x} + \sqrt{7 - 2 \sin x}) . \tan x} =

\lim_{x \to π} \frac{1}{\sqrt{7 + 2 \sin x} + \sqrt{7 - 2 \sin x}} \times \lim_{x \to π} \frac{4 \sin x}{\tan x}

= \frac{1}{2 \sqrt{7}} \times \lim_{x \to π} 4 \sin x (\frac{\cos x}{\sin x})

= \frac{1}{2 \sqrt{7}} \times (- 4) = - \frac{2}{\sqrt{7}} .

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