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$\lim\limits_{x\to 1}(\log_22x)^{\large\log_x5}$ is

$(a)\;5/2\qquad(b)\;e^{\large\log_25}\qquad(c)\;\large\frac{\log 5}{\log 2}$$\qquad(d)\;e^{\large\log_52}$

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$\lim\limits_{x\to 1}(\log_22x)^{\large\log_x5}=\lim\limits_{x\to 1}(\log_2 2+\log_2x)^{\large\log_x 5}$
$\lim\limits_{x\to 1}(1+\log_2x)^{\Large\frac{1}{\log_5x}}$
$\Rightarrow \lim\limits_{x\to 1}\big[(1+\log_2 x)^{1/\log_2x}\big]^{\Large\frac{\log_2x}{\log_5x}}$
$\Rightarrow e^{\lim\limits_{x\to 1}\Large\frac{\log_2x}{\log_5x}}$
$\Rightarrow e^{\large\lim\limits_{x\to 1}\log_25}$
$\Rightarrow e^{\large\log_25}$
Hence (b) is the correct answer.
answered Jan 2, 2014 by sreemathi.v
 
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