Evaluate: `("lim")_(xvec0)((a^x+b^x+c^x)/3);(a , b , c >0)`

Explanation

We have
`underset(xto0)lim((a^(x)+b^(x)+c^(x))/(3))^(2//x)=e^(underset(xto0)lim((a^(x)+b^(x)+c^(x))/(3)-1).2/x`
`=e^(2/3underset(xto0)lim((a^(x)+b^(x)+c^(x)-3)/(x))`
`=e^(2/3underset(xto0)lim((a^(x)-1)/(x)+(b^(x)-1)/(x)+(c^(x)-1)/(x))`
`=e^(2/3{underset(xto0)lima^(x-1)/(x)+underset(xto0)lim(b^(x)-1)/(x)+underset(xto0)lim(c^(x)-1)/(x)}}`
`=e^((2//3){"In "a+"In "b+"In "c}}`
`=e^((2//3)"In"(abc))`
`=e^("In"(abc)^(2//3))`
`=(abc)^(2//3)`