GATE AG 2020 • Q25 NAT • 1 Mark

Section 1: Engineering Mathematics

Calculus

Subtopic: Sequences and series – infinite series, tests for convergence

Question Statement:

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

Official Answer Key:

2 to 2

Step-by-Step Solution & Explanation

Series: eˣ−e⁻ˣ = 2x + x³/3 + O(x⁵) ⇒ numerator ≈ x³/3. sin x = x − x³/6 + O(x⁵) ⇒ denominator (x−sinx) ≈ x³/6. Limit = (x³/3)/(x³/6) = 2.

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