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Introduction to limits

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

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Solution

$\begin{array}{l} lim_{x \to \frac{π}{2}} \frac{4 \sqrt{2} \cdot 2 \sin 2 x \cos x}{2 \sin 2 x \sin \frac{3 x}{2} + (\cos \frac{5 x}{2} - \cos \frac{3 x}{2}) - \sqrt{2} (1 + \cos 2 x)} \\ lim_{x \to \frac{π}{2}} \frac{16 \sqrt{2} \sin x \cos^{2} x}{2 \sin 2 x (\sin \frac{3 x}{2} - \sin \frac{x}{2}) - 2 \sqrt{2} \cos^{2} x} \end{array}$
$\begin{array}{l} lim_{x \to \frac{π}{2}} \frac{16 \sqrt{2} \sin x \cos^{2} x}{4 \sin x \cos x (2 \cos x \cdot \sin \frac{x}{2}) - 2 \sqrt{2} \cos^{2} x} \\ lim_{x \to \frac{π}{2}} \frac{16 \sqrt{2} \sin x}{8 \sin x \cdot \sin \frac{x}{2} - 2 \sqrt{2}} = 8 \end{array}$

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