QJEEM21Ma300f: JEE Main 2021 Mathematics - Class 12 JEEM21Ma300f Solution: Let f : R R be a continuous function. Then is equal to :

Question

Verified Solution • 2025–26 Session moderate

Let f : R

\to

R be a continuous function. Then

\mathop {\lim }\limits_{x \to {\pi \over 4}} {{{\pi \over 4}\int\limits_2^{{{\sec }^2}x} {f(x)\,dx} } \over {{x^2} - {{{\pi ^2}} \over {16}}}}

is equal to :

Accepted Answer

\mathop {\lim }\limits_{x \to {\pi \over 4}} {{{\pi \over 4}\int\limits_2^{{{\sec }^2}x} {f(x)\,dx} } \over {{x^2} - {{{\pi ^2}} \over {16}}}}

=

\mathop {\lim }\limits_{x \to {\pi \over 4}} {\pi \over 4}.{{\left[ {f({{\sec }^2}x).2\sec x.\sec x\tan x} \right]} \over {2x}}

=

\mathop {\lim }\limits_{x \to {\pi \over 4}} {\pi \over 4}f({\sec ^2}x){\sec ^3}x.{{\sin x} \over x}

=

{\pi \over 4}f(2).{\left( {\sqrt 2 } \right)^3}.{1 \over {\sqrt 2 }} \times {4 \over \pi }

= 2f (2)

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