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JEE Main & Advanced Mathematics Functions Question Bank Critical Thinking

  • question_answer
    Let \[f(x)=\frac{1-\tan x}{4x-\pi },\ x\ne \frac{\pi }{4},\ \ x\in \left[ 0,\frac{\pi }{2} \right]\], If \[f(x)\]is continuous in \[\left[ 0,\frac{\pi }{2} \right]\], then \[f\left( \frac{\pi }{4} \right)\]is [AIEEE 2004]

    A) -1

    B) \[\frac{1}{2}\]

    C) \[-\frac{1}{2}\]

    D) 1

    Correct Answer: C

    Solution :

    • \[\underset{x\to \frac{\pi }{4}}{\mathop{\lim }}\,f(x)=\underset{x\to \frac{\pi }{4}}{\mathop{\lim }}\,\frac{1-\tan x}{4x-\pi },\,\,\,\left[ \frac{0}{0}\text{form} \right]\]                   
    • \[\underset{x\to \frac{\pi }{4}}{\mathop{\lim }}\,\frac{-{{\sec }^{2}}x}{4}=\frac{-2}{4}=\frac{-1}{2}\].                   
    • \ For f(x) to be continuous at \[x=\frac{\pi }{4},\ f\left( \frac{\pi }{4} \right)=\frac{-1}{2}\]


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