The value of the limit lim x 2 4 2 ( sin 3 x + sin x ) ( 2 sin 2 x sin 3 x 2 + cos 5 x 2 ) - ( 2 + 2 cos 2 x + cos 3 x 2 ) is ____________ [2020]

Question

The value of the limit $\lim_{x \to \frac{π}{2}} \frac{4 \sqrt{2} (\sin 3 x + \sin x)}{(2 \sin 2 x \sin \frac{3 x}{2} + \cos \frac{5 x}{2}) - (\sqrt{2} + \sqrt{2} \cos 2 x + \cos \frac{3 x}{2})}$ is ____________ [2020]

Smart Achievers · Accepted Answer

(8)

$\lim_{x \to \frac{π}{2}} \frac{4 \sqrt{2} \cdot 2 \sin 2 x \cos x}{2 \sin 2 x \sin \frac{3 x}{2} + (\cos \frac{5 x}{2} - \cos \frac{3 x}{2}) - \sqrt{2} (1 + \cos 2 x)}$

$= \lim_{x \to \frac{π}{2}} \frac{8 \sqrt{2} \cdot 2 \sin x \cos x \cos x}{2 \sin 2 x \sin \frac{3 x}{2} - 2 \sin 2 x \sin \frac{x}{2} - 2 \sqrt{2} \cos^{2} x}$

$= \lim_{x \to \frac{π}{2}} \frac{16 \sqrt{2} \sin x \cos^{2} x}{2 \sin 2 x (\sin \frac{3 x}{2} - \sin \frac{x}{2}) - 2 \sqrt{2} \cos^{2} x}$

$= \lim_{x \to \frac{π}{2}} \frac{16 \sqrt{2} \sin x \cos^{2} x}{4 \sin x \cos x (2 \cos x . \sin \frac{x}{2}) - 2 \sqrt{2} \cos^{2} x}$

$= \frac{16 \sqrt{2} \sin x \cos^{2} x}{2 \cos^{2} x (4 \sin x \sin \frac{x}{2} - \sqrt{2})}$

$= \lim_{x \to \frac{π}{2}} \frac{8 \sqrt{2} \sin x}{4 \sin x . \sin \frac{x}{2} - \sqrt{2}} = 8$

Solution

Q.
The value of the limit $\lim_{x \to \frac{π}{2}} \frac{4 \sqrt{2} (\sin 3 x + \sin x)}{(2 \sin 2 x \sin \frac{3 x}{2} + \cos \frac{5 x}{2}) - (\sqrt{2} + \sqrt{2} \cos 2 x + \cos \frac{3 x}{2})}$ is ____________ [2020]

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Solution

Q. The value of the limit limx→π242(sin3x+sinx)(2sin2xsin3x2+cos5x2)-(2+2cos2x+cos3x2) is ____________ [2020]

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Q.
The value of the limit $\lim_{x \to \frac{π}{2}} \frac{4 \sqrt{2} (\sin 3 x + \sin x)}{(2 \sin 2 x \sin \frac{3 x}{2} + \cos \frac{5 x}{2}) - (\sqrt{2} + \sqrt{2} \cos 2 x + \cos \frac{3 x}{2})}$ is ____________ [2020]