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GATE 2021 ME (ME1) – Question 12

Engineering Mathematics · Calculus: Limit, continuity, differentiability, mean value theorems and indeterminate forms · 1 mark · Multiple choice

The value of $\lim_{x\to0}\frac{1-\cos x}{x^2}$ is

  1. $1/4$
  2. $1/3$
  3. $1/2$
  4. 1

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Correct answer: (C) $1/2$

Explanation

Use $1-\cos x=2\sin^2(x/2)$. The ratio becomes $\tfrac12[\sin(x/2)/(x/2)]^2$.

The standard sine limit is 1, so the result is **1/2 (C)**. The quadratic Taylor term gives the same limit.