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JEE Main & Advanced Mathematics Functions Question Bank Limits

  • question_answer
    Let \[f:R\to R\]be a differentiable function having \[f(2)=6,f'(2)=\left( \frac{1}{48} \right).\] Then \[\underset{x\to 2}{\mathop{\lim }}\,\int\limits_{6}^{f(x)}{\frac{4{{t}^{3}}}{x-2}}\]dt equals [AIEEE 2005]

    A)                 12

    B)                 18

    C)                 24

    D)                 36

    Correct Answer: B

    Solution :

                       \[\underset{x\to 2}{\mathop{\lim }}\,\frac{\int\limits_{6}^{f(x)}{4{{t}^{3}}dt}}{x-2}\,\,(0/0\,\text{form})=\underset{x\to 2}{\mathop{\lim }}\,\frac{4{{(f(x))}^{3}}\times f'(x)}{1}\]                 \[=4{{(f(2))}^{3}}\times f'(2)=18\].


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