Question

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

• 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

Answer: (A)

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

$\underset{x\to \frac{\pi }{4}}{lim}\frac{\stackrel{{\sec }^{2}x}{\underset{2}{\int }}f\left(t\right)dt}{x^2-\frac{{\pi }^2}{16}}$ $\left(\frac{0}{0}\right)$

form.

Now applying L'Hospital's rule we get,

$=\underset{x\to \frac{\pi }{4}}{lim}\frac{f\left({\sec }^{2}x\right).2{\sec }^{2}x.\tan x-f\left(2\right).0}{2x}$

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

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