Question

$\underset{x\to \pi /4}{lim}\frac{{\int }_2^{{\sec }^{2}x}f\left(t\right)dt}{x^2-\frac{{\pi }^2}{16}}$ 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 $\frac{8}{\pi }f\left(2\right)$

$\frac{0}{0}$ form :- applying L-Hospital rule $\Rightarrow$

$\underset{x\to \frac{\pi }{4}}{lim}\frac{f\left({\sec }^{2}x\right)2{\sec }^{2}x\tan x}{2x}$

$=\frac{f\left(2\right).2\left(2\right)\left(1\right)}{2\left(\frac{\pi }{4}\right)}$

$=\frac{8}{\pi }f\left(2\right)$