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# $\lim\limits_{x\to 0}\big(\large\frac{a^x+b^x+c^x}{3}\big)^{\Large\frac{1}{x}}$ is

$(a)\;abc\qquad(b)\;\sqrt{abc}\qquad(c)\;(abc)^{\Large\frac{1}{3}}\qquad(d)\;None\;of\;these$

Can you answer this question?

$\lim\limits_{x\to 0}\big(\large\frac{a^x+b^x+c^x}{3}\big)^{1/x}=e^{\large\lim\limits_{x\to 0}\large\frac{1}{x}\big(\Large\frac{a^x+b^x+c^x}{3}\normalsize -1\big)}$
$\Rightarrow e^{\large\lim\limits_{x\to 0}\big(\Large\frac{a^x-1}{x}+\frac{b^x-1}{x}+\frac{c^x-1}{x}\big)\frac{1}{3}}$
$\Rightarrow e^{\Large\frac{\log a+\log b+\log c}{3}}$
$\Rightarrow (abc)^{\large\frac{1}{3}}$
Hence (c) is the correct answer.
answered Dec 23, 2013