$\lim _{x \rightarrow 0} \operatorname{cosec} x\left(\sqrt{2 \cos ^{2} x+3 \cos x}-\sqrt{\cos ^{2} x+\sin x+4}\right)$ is

JEE Main · 2025 · 24 Jan 2025 · Shift 1 · Limits

lim_{x \to 0} cosec x (2 cos^{2} x + 3 cos x - cos^{2} x + sin x + 4)

Answer: (D)

Near

x = 0

, use

cos x \approx 1 - \frac{x ^{2}}{2}

and

sin x \approx x

2 cos^{2} x + 3 cos x \approx 2 (1 - x^{2}) + 3 (1 - \frac{x ^{2}}{2}) = 5 - \frac{7}{2} x^{2}, 2 cos^{2} x + 3 cos x \approx 5 (1 - \frac{7}{20} x^{2}) .

cos^{2} x + sin x + 4 \approx (1 - x^{2}) + x + 4 = 5 + x - x^{2}, cos^{2} x + sin x + 4 \approx 5 (1 + \frac{x}{10})

keeping only the linear term, since it dominates the quadratic term as

x \to 0

.

Subtracting,

2 cos^{2} x + 3 cos x - cos^{2} x + sin x + 4 \approx - 5 \cdot \frac{x}{10} .

Multiplying by

cosec x \approx \frac{1}{x}

x \to 0 lim cosec x (2 cos^{2} x + 3 cos x - cos^{2} x + sin x + 4) = - \frac{5}{10} = - \frac{1}{2 5} .

JEE Main · 2025 · 24 Jan 2025 · Shift 1 · Single correct option

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