Q31 / 66 · JEE Advanced 2007 (Paper 1) · Mathematics

$\lim _{x \rightarrow \frac{\pi}{4}} \frac{\int_2^{\sec ^2 x} f(t) d t}{x^2-\frac{\pi^2}{16}}$ equals

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

Solution

$\lim _{x \rightarrow \frac{x}{4}} \frac{\int_2^{\sec ^2 x} f(t) d t}{x^2-\frac{\pi^2}{16}}$ $\left[\therefore \frac{0}{0}\right.$ form $]$ Let $$ \begin{array}{ll} \text { Let } & L=\lim _{x \rightarrow \frac{x}{4}} \frac{f\left(\sec ^2 x\right) 2 \sec x \sec x \tan x}{2 x} \\ \therefore & L=\frac{2 f(2)}{\pi / 4}=\frac{8 f(2)}{\pi} \end{array} $$

Hence, the correct answer is option: 'A'.

Relevant Questions