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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 30

  • question_answer
    \[\underset{x\to \frac{\pi }{4}}{\mathop{\lim }}\,\frac{\int\limits_{2}^{{{\sec }^{2}}x}{f(t)dt}}{{{x}^{2}}-\frac{{{\pi }^{2}}}{16}}\]equals

    A) \[\frac{8}{\pi }f(2)\]       

    B) \[\frac{2}{\pi }f(2)\]

    C) \[\frac{2}{\pi }f\left( \frac{1}{2} \right)\]  

    D) \[4f(2)\]

    Correct Answer: A

    Solution :

    [a]: Let\[L=\underset{x\to \frac{\pi }{4}}{\mathop{\lim }}\,\frac{\int\limits_{2}^{{{\sec }^{2}}x}{f(t)dt}}{{{x}^{2}}-\frac{{{\pi }^{2}}}{16}}\] Applying L? Hospital rule \[L=\underset{x\to \frac{\pi }{4}}{\mathop{\lim }}\,\frac{f(se{{c}^{2}}x).2se{{c}^{2}}x\tan x-0}{2x}\] \[L=\frac{f(2).4}{\pi /2}=\frac{8f(2)}{\pi }.\]


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