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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 17

  • question_answer
    If \[f(x)\] is a differentiable function such that \[f'(1)=4\] and \[f'(4)=\frac{1}{2},\] then value of  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{f({{x}^{2}}+x+1)-f(1)}{f({{x}^{4}}-{{x}^{2}}+2x+4)-f(4)}\] is

    A) 8                      

    B)        16                     

    C) 4       

    D)        Does not exist

    Correct Answer: C

    Solution :

    [c] \[\underset{x\to 0}{\mathop{\lim }}\,\frac{f({{x}^{2}}+x+1)-f(1)}{f({{x}^{4}}-{{x}^{2}}+2x+4)-f(4)}\]   \[\left( \frac{0}{0}from \right)\] \[=\underset{x\to 0}{\mathop{\lim }}\,\frac{(2x+1)f'({{x}^{2}}+x+1)}{(4{{x}^{3}}-2x+2)f'({{x}^{4}}-{{x}^{2}}+2x+4)}=\frac{f'(1)}{2f'(4)}=4\] (Applying L' Hospital's Rule)             


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