Question

$\underset{x\to \frac{\pi }{4}}{\lim }\frac{\underset{2}{\overset{{\sec }^{2}x}{\int }}f(t)dt}{x^2-\frac{{\pi }^2}{16}}$ equals

• A. $\frac{8}{\pi }f$
• B. $\frac{2}{\pi }f$
• C. $\frac{2}{\pi }f\left(\frac{1}{2}\right)$
• D. $4f(2)$

Answer 1

Answer: (A)

$\underset{x\to \frac{\pi }{4}}{\lim }\frac{\underset{2}{\overset{{\sec }^{2}x}{\int }}f(t)dt}{x^2-\frac{{\pi }^2}{16}}[\frac{0}{0}$ form]
$=\underset{x\to \pi /4}{\lim }\frac{f\left({\sec }^{2}x\right)2\sec x\sec x\tan x}{2x}$
[using L' Hospital's rule]
$=\frac{2f(2)}{\pi /4}=\frac{8}{\pi }f(2)$