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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→0) (((a^x +b^x +c^x )/3))^(1/x)
limx→0(ax+bx+cx3)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) .
limx→0(1+ax−1+bx−1+cx−13)1/x=limx→0{[1+f(a,b,c,x)]1f(a,b,c,x)}f(a,b,c,x)x=elna+lnb+lnc3=(abc)1/3.

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