Q. $\lim_{x \rightarrow \frac{\pi}{4}}\frac{\int_{2}^{\sec^{2}x}{f(t)dt}}{x^{2} - \frac{\pi^{2}}{16}}$ equals
A) $\frac{8}{\pi}f(2)$
B) $\frac{2}{\pi}f(2)$
C) $\frac{2}{\pi}f\left( \frac{1}{2} \right)$
D) $4f(2)$
Explanation
(a)
$\lim_{x \rightarrow \frac{\pi}{4}}\frac{\int_{2}^{\sec^{2}x}{f(t)dt}}{x^{2} - \frac{\pi^{2}}{16}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left\lbrack \frac{0}{0}\ from \right\rbrack$
$= \lim_{x \rightarrow \frac{\pi}{4}}\frac{f\left( \sec^{2}x \right)2\sec{x\sec{x\tan x}}}{2x}$
$\therefore\ \ \ \ \ \ \ \ L = \frac{2f(2)}{\pi/4} = \frac{8f(2)}{\pi}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.