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

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

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