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JEE Advanced 2007Paper 1

Exam: JEE Advanced
Year: 2007
Paper: Paper 1
Subject: Mathematics
Chapter: Limits

JEE Advanced 2007 · Paper 1 · Mathematics · Limits

_ x arrow (π)/(4) ∫_2^ ^2 x f(t) d t x^2-(π^2)/(16) equals

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

equals

Options

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

Correct answer

A. $\frac{8}{\pi} f(2)$

Step-by-step 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}

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