Question

Let $f:R\to R$ be a continuous function. Then $\underset{x\to \frac{\pi }{4}}{\lim }\frac{\frac{\pi }{4}\underset{2}{\overset{{\sec }^{2}x}{\int }}f(x)dx}{x^2-\frac{{\pi }^2}{16}}$ is equal to:

• A. $f(2)$
• B. $2f(2)$
• C. $2f(\sqrt{2})$
• D. $4f(2)$

Answer 1

Answer: (B)

$\underset{x\to \frac{\pi }{4}}{\lim }\frac{\frac{\pi }{4}\underset{2}{\overset{{\sec }^{2}x}{\int }}f(x)dx}{x^2-\frac{{\pi }^2}{16}}$
$\underset{x\to \frac{\pi }{4}}{\lim }\frac{\pi }{4}\cdot \frac{f\left({\sec }^{2}x\right)\cdot 2\sec x\cdot \sec x\tan x]}{2x}$
$\underset{x\to \frac{\pi }{4}}{\lim }\frac{\pi }{4}f\left({\sec }^{2}x\right)\cdot {\sec }^{3}x\cdot \frac{\sin x}{x}$
$\frac{\pi }{4}f(2)\cdot (\sqrt{2}{)}^3\cdot \frac{1}{\sqrt{2}}\times \frac{4}{\pi }$
$\Rightarrow 2f(2)$