GATE AG 2025 • Q37 MCQ • 2 Marks

Section 1: Engineering Mathematics

Calculus

Subtopic: Limit, continuity and differentiability

Question Statement:

Find $\lim_{x \to 0} \frac{x\cos x - \sin x}{x^2 \sin x}$:

Options:

  • (A) −1/3
  • (B) 1/3
  • (C) 3
  • (D) −3

Official Answer Key:

A

Step-by-Step Solution & Explanation

Using series expansions: cos x ≈ 1−x²/2, sin x ≈ x−x³/6. Numerator ≈ x(1−x²/2) − (x−x³/6) = x − x³/2 − x + x³/6 = −x³/3. Denominator ≈ x²·x = x³. Limit = −1/3 (confirmed symbolically).

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