GATE AG 2023 • Q12
MCQ • 1 Mark
Section 1: Engineering Mathematics
Calculus
Subtopic: Total differentiation
Question Statement:
$\lim_{x \to 0} \frac{\sqrt{1+x} - 1}{x}$ is equal to:
Options:
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(A) 0
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(B) 1/2
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(C) 1
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(D) 2
Step-by-Step Solution & Explanation
Rationalising: [√(1+x)−1]/x × [√(1+x)+1]/[√(1+x)+1] = [(1+x)−1]/[x(√(1+x)+1)] = x/[x(√(1+x)+1)] = 1/[√(1+x)+1]. As x→0, this → 1/(1+1) = 1/2. (Equivalently, by L'Hopital's rule: d/dx[√(1+x)−1]/d/dx[x] = [1/(2√(1+x))]/1 → 1/2 at x=0.) Answer: (B) 1/2.
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