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JEE Main 2021 · Mathematics · Limits · medium

Lim x→0(e^x^2-cos x)/(x^2 is equal to

Single correct option

lim⁡x→0ex2−cos⁡xx2\displaystyle\lim_{x\to0}\frac{e^{x^2}-\cos x}{x^2} is equal to
  1. A. 12\dfrac12
  2. B. 11
  3. C. 32\dfrac32
  4. D. 22
Show answer and solution

Answer: (C) 32\dfrac32

Use series: ex2=1+x2+…e^{x^2}=1+x^2+\dots and cos⁡x=1−x22+…\cos x=1-\dfrac{x^2}{2}+\dots. So ex2−cos⁡x=3x22+O(x4)e^{x^2}-\cos x=\dfrac{3x^2}{2}+O(x^4) and the limit is 32\dfrac32. Equivalently split as ex2−1x2+1−cos⁡xx2→1+12\dfrac{e^{x^2}-1}{x^2}+\dfrac{1-\cos x}{x^2}\to1+\dfrac12.
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Original question written in the JEE Main 2021 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake