GATE AG 2021 • Q12 MCQ • 1 Mark

Section 1: Engineering Mathematics

Calculus

Subtopic: Limit, continuity and differentiability

Question Statement:

If x is an integer with x > 1, the solution of lim(x→∞) (1/x² + 2/x² + 3/x² + ⋯ + (x−1)/x² + 1/x) is

Options:

  • (A) Zero
  • (B) 0.5
  • (C) 1.0
  • (D) ∞

Official Answer Key:

B

Step-by-Step Solution & Explanation

The sum 1+2+⋯+(x−1) = x(x−1)/2, so the series (excluding the last term 1/x) equals [x(x−1)/2]/x² = (x−1)/(2x). Adding the extra 1/x term: total = (x−1)/(2x) + 1/x = (x−1+2)/(2x) = (x+1)/(2x). As x → ∞, (x+1)/(2x) → 1/2 = 0.5, matching (B).

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