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JEE Main & Advanced Mathematics Functions Question Bank Critical Thinking

  • question_answer
    Let \[f(x)\]be defined for all \[x>0\]and be continuous. Let \[f(x)\]satisfy \[f\left( \frac{x}{y} \right)=f(x)-f(y)\]for all x, y and \[f(e)=1,\]then [IIT 1995]

    A) \[f(x)=\ln x\]

    B) \[f(x)\]is bounded

    C) \[f\left( \frac{1}{x} \right)\to 0\]as\[x\to 0\]

    D) \[x\,f(x)\to 1\]as \[x\to 0\]

    Correct Answer: A

    Solution :

    • Let \[f(x)=\]ln \[(x),\,\,x>0\] \[f(x)=\] ln\[(x)\] is a continuous function of x for every positive value of x.                   
    • \[0<x<1\]ln \[\left( \frac{x}{y} \right)=\]ln (x)? ln (y)=f(x)? f(y).


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