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$\;\lim \limits_{x \to 0}\Large\frac{a^{x}-b^{x}}{c^{x}-d^{x}}$ is

\[\begin{array}{1 1}(1)\infty&(2)0\\(3)\log\frac{ab}{cd}&(4)\frac{\log(a/b)}{\log(c/d)}\end{array}\]

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$\lim \limits_{x \to 0} \large\frac{a^x-^x}{c^x -d^x} =\lim \limits _{x \to 0} \large\frac{a^x \log a -b^{x} \log b}{c^x \log c- d^x \log d}$
$\qquad = \large\frac{a^0 \log a -b^0 \log b}{c^0 \log c -d^0 \log d}$
$\qquad= \large\frac{\log a -\log b}{\log c -\log d}$
$\qquad=\large\frac{ \log a/b}{\log c/d}$
Hence 4 is the correct answer.
answered May 19, 2014 by meena.p
 

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