lim_(x→0) (((a^x +b^x +c^x )/3))^(1/x)


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Question Number 46563 by Necxx last updated on 28/Oct/18

$\lim_{x \to 0} {(\frac{a^{x} + b^{x} + c^{x}}{3})}^{1 / x}$
	Answered by ajfour last updated on 28/Oct/18
	$lim_(x→0) (1+((a^x −1+b^x −1+c^x −1)/3))^(1/x) =lim_(x→0) {[1+f(a,b,c,x)]^(1/(f(a,b,c,x))) }^((f(a,b,c,x))/x) = e^(((ln a+ln b+ln c)/3) ) = (abc)^(1/3) .$
	$\lim_{x \to 0} {(1 + \frac{a^{x} - 1 + b^{x} - 1 + c^{x} - 1}{3})}^{1 / x}$ $= \lim_{x \to 0} {{[1 + f (a, b, c, x)]}^{\frac{1}{f (a, b, c, x)}}}^{\frac{f (a, b, c, x)}{x}}$ $= e^{\frac{\ln a + \ln b + \ln c}{3}} = {(a b c)}^{1 / 3} .$
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