Question

$\underset{x\to \pi /4}{lim}\frac{{\int }_2^{{\sec }^{2}x}f\left(t\right)dt}{x^2-{\pi }^2/16}$ equals

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

Answer 1

The correct option is A $\frac{8}{\pi }f\left(2\right)$

Required limit is of the form $\frac{0}{0}$ so it is equal to
$\underset{x\to \pi /4}{lim}\frac{2\sec x\sec x\tan xf\left({\sec }^{2}x\right)}{2x}$ (use Property 17)
$=\underset{x\to \pi /4}{lim}\frac{{\sec }^{2}x\tan xf\left({\sec }^{2}x\right)}{x}$
$=\underset{x\to \pi /4}{lim}\frac{{\sec }^2\left(\pi /4\right)\tan \pi /4f\left({\sec }^2\pi /4\right)}{\frac{\pi }{4}}$

$=\frac{\left(\sqrt{2}{)}^{2}1f\right(\sqrt{2}{)}^2}{\pi /4}=\frac{8}{\pi }f\left(2\right)$