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JEE Main 201226 MayOnline
Exam
JEE Main
Year
2012
Date
26 May
Mode
Online
Subject
Mathematics
Chapter
Limits

JEE Main 2012 · 26 May · Online · Mathematics · Limits

_ x arrow 0 ( (π ^2 x))/(x^2) equals

\lim _{x \rightarrow 0} \frac{\sin \left(\pi \cos ^2 x\right)}{x^2} equals

Options

  1. A−π-\pi
  2. B1
  3. C−1-1
  4. Dπ\pi

Correct answer

D. π\pi

Step-by-step solution

Consider lim⁡x→0sin⁡(πcos⁡2x)x2=lim⁡x→0sin⁡(π−πsin⁡2x)x2=lim⁡x→0sin⁡(πsin⁡2x)πsin⁡2x×(πsin⁡2x)x2=π\begin{aligned}& \lim _{x \rightarrow 0} \frac{\sin \left(\pi \cos ^2 x\right)}{x^2}\\ & =\lim _{x \rightarrow 0} \frac{\sin \left(\pi-\pi \sin ^2 x\right)}{x^2}\\ & =\lim _{x \rightarrow 0} \frac{\sin \left(\pi \sin ^2 x\right)}{\pi \sin ^2 x} \times \frac{\left(\pi \sin ^2 x\right)}{x^2}=\pi \end{aligned}
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