JEE Advanced · Mathematics · 16. Limits
\(\lim _{x \rightarrow \frac{\pi}{4}} \frac{\int_2^{\sec ^2 x} f(t) d t}{x^2-\frac{\pi^2}{16}}\) equals
- A
\(\frac{8}{\pi} f(2)\)
- B
\(\frac{2}{\pi} f(2)\)
- C
\(\frac{2}{\pi} f\left(\frac{1}{2}\right)\)
- D
\(4 f(2)\)
Answer & Solution
Correct Answer
(A)
\(\frac{8}{\pi} f(2)\)
Step-by-step Solution
Detailed explanation
\(\lim _{x \rightarrow \frac{x}{4}} \frac{\int_2^{\sec ^2 x} f(t) d t}{x^2-\frac{\pi^2}{16}}\) \(\left[\therefore \frac{0}{0}\right.\) form \(]\)
Let
\[
\begin{array}{ll}
\text { Let } & L=\lim _{x \rightarrow \frac{x}{4}} \frac{f\left(\sec ^2 x\right) 2 \sec x \sec x \tan x}{2 x} \\
\therefore & L=\frac{2 f(2)}{\pi / 4}=\frac{8 f(2)}{\pi}
\end{array}
\]
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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