The value of limx→π/4 22−(cos⁡x+sin⁡x)31−sin⁡2x,is

A
$\frac{3}{\sqrt{2}}$
B
$\frac{\sqrt{2}}{3}$
C
$\frac{1}{\sqrt{2}}$
D
$\sqrt{2}$

Solution:

We have,

$\begin{array}{l} lim_{x \to π / 4} \frac{2 \sqrt{2} - (\cos x + \sin x)^{3}}{1 - \sin 2 x} \\ = lim_{x \to π / 4} \frac{2^{3 / 2} - {[(\cos x + \sin x)^{2}]}^{3 / 2}}{2 - (1 + \sin 2 x)} \\ = lim_{x \to π / 4} \frac{2^{3 / 2} - (1 + \sin 2 x)^{3 / 2}}{2 - (1 + \sin 2 x)} \end{array}$

$= lim_{y \to 2} \frac{y^{3 / 2} - 2^{3 / 2}}{y - 2},$ where $y = 1 + \sin 2 x$

$= \frac{3}{2} (2)^{3 / 2 - 1} = \frac{3}{2} \times \sqrt{2} = \frac{3}{\sqrt{2}}$

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