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NDA 2016 Maths · Mathematics · Limits · hard

Lim x→0(1-cos x)/(x^2 is equal to

Single correct option

lim⁡x→01−cos⁡xx2\displaystyle\lim_{x\to0}\frac{1-\cos x}{x^2} is equal to
  1. A. 11
  2. B. 00
  3. C. 22
  4. D. 12\dfrac12
Show answer and solution

Answer: (D) 12\dfrac12

Use 1−cos⁡x=2sin⁡2(x/2)1-\cos x=2\sin^2(x/2). Then the ratio is 2sin⁡2(x/2)x2=12(sin⁡(x/2)x/2)2→12\dfrac{2\sin^2(x/2)}{x^2}=\dfrac12\left(\dfrac{\sin(x/2)}{x/2}\right)^2\to\dfrac12. The limit is not 11 because the denominator is x2x^2, not xx.
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Original question written in the NDA 2016 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake