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JEE Main & Advanced Mathematics Functions Question Bank Critical Thinking

  • question_answer
    If \[\underset{x\to a}{\mathop{\lim }}\,\frac{{{a}^{x}}-{{x}^{a}}}{{{x}^{x}}-{{a}^{a}}}=-1\], then                              [EAMCET 2003]

    A) \[a=1\]

    B) \[a=0\]

    C) \[a=e\]

    D) None of these

    Correct Answer: A

    Solution :

    • Using L-Hospital's rule, we get           
    • \[-1=\underset{x\to a}{\mathop{\lim }}\,\,\,\frac{{{a}^{x}}-{{x}^{a}}}{{{x}^{x}}-{{a}^{a}}}=\underset{x\to a}{\mathop{\lim }}\,\,\frac{{{a}^{x}}{{\log }_{e}}a-a{{x}^{a-1}}}{{{x}^{x}}+{{a}^{a}}{{\log }_{e}}a}\]           
    • \[\Rightarrow \,\,-1=\frac{{{a}^{a}}{{\log }_{e}}a-a\,.\,{{a}^{a-1}}}{{{a}^{a}}+{{a}^{a}}{{\log }_{e}}a}=\frac{{{\log }_{e}}a-1}{{{\log }_{e}}a+1}\]     .....(i)     
    • Now (i) is satisfied only when \[a=1.\]


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