lim x 0 x ( 2 cos 2 x + 3 cos x – cos 2 x + sin x + 4 ) is : [2025]

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Solution

Q.
$\lim_{x \to 0} \cos e c x (\sqrt{2 \cos^{2} x + 3 \cos x} - \sqrt{\cos^{2} x + \sin x + 4})$ is : [2025]

1 $- \frac{1}{2 \sqrt{5}}$

2 $\frac{1}{\sqrt{15}}$

3 0

4 $\frac{1}{2 \sqrt{5}}$

Ans.
(1)

$\lim_{x \to 0} \cos e c x (\sqrt{2 \cos^{2} x + 3 \cos x} - \sqrt{\cos^{2} x + \sin x + 4})$

$= \lim_{x \to 0} \frac{1}{\sin x} [\frac{2 \cos^{2} x + 3 \cos x - \cos^{2} x - \sin x - 4}{\sqrt{2 \cos^{2} x + 3 \cos x} + \sqrt{\cos^{2} x + \sin x + 4}}]$

$= \lim_{x \to 0} \frac{1}{\sin x} [\frac{\cos^{2} x + 3 \cos x - \sin x - 4}{\sqrt{2 \cos^{2} x + 3 \cos x} + \sqrt{\cos^{2} x + \sin x + 4}}]$

$= \lim_{x \to 0} [\frac{(\cos x + 4) (\cos x - 1) - \sin x}{\sin x}] \times \lim_{x \to 0} [\frac{1}{\sqrt{2 \cos^{2} x + 3 \cos x} + \sqrt{\cos^{2} x + \sin x + 4}}]$

$= \frac{1}{2 \sqrt{5}} \lim_{x \to 0} [\frac{(\cos x + 4) (- 2 \sin^{2} \frac{x}{2}) - 2 \sin \frac{x}{2} \cos \frac{x}{2})}{2 \sin \frac{x}{2} \cos \frac{x}{2}}]$

$= \frac{1}{2 \sqrt{5}} \lim_{x \to 0} [\frac{(- \sin \frac{x}{2}) (\cos x + 4) - \cos \frac{x}{2}}{\cos \frac{x}{2}}]$

$= \frac{1}{2 \sqrt{5}} [- 1] = \frac{- 1}{2 \sqrt{5}}$ .

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