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JCECE Engineering JCECE Engineering Solved Paper-2014

  • question_answer
    \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{2}^{x}}-1}{{{(1+x)}^{1/2}}-1}\]is equal to

    A) \[\log 2\]                            

    B) \[\log 4\]

    C) \[\log \sqrt{2}\]                               

    D)  None of these

    Correct Answer: B

    Solution :

    \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{2}^{x}}-1}{{{(1+x)}^{1/2}}-1}=\underset{x\to 0}{\mathop{\lim }}\,\frac{{{2}^{x}}\log 2}{\frac{1}{2}{{(1+x)}^{-1/2}}}\] \[=2\log 2\left[ \because \underset{x\to a}{\mathop{\lim }}\,\frac{f(x)}{g(x)}=\underset{x\to a}{\mathop{\lim }}\,\frac{f'(x)}{g'(x)} \right]\]                 \[=\log 4\]


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