Let` a_(1)gta_(2)gta_(3)gt...gta_(n)gt1.`
`p_(1)gtp_(2)gtp_(3)gt...gtp_(n)gt0" such that "p_(1)+p_(2)+p_(3)+...+p_(n)=1.`
Also, `F(x)=(p_(1)a_(1)^(x)+...p_(n)a_(n)^(x))^(1//x)`.
`lim_(xtooo) F(x)" equals "`

Options

A. `{:(a,b,c,d),(s,r,q,p):}`
B. `{:(a,b,c,d),(q,p,s,p):}`
C. `{:(a,b,c,d),(s,r,p,q):}`
D. `{:(a,b,c,d),(p,p,q,r):}`

Answer

C

Explanation

`underset(xto0^(+))limF(x)=underset(xto0^(+))lim(p_(1)a_(1)^(x)+p_(2)a_(2)^(x)+...+p_(n)a_(n)^(x))^(1//x)" "(1^(oo)" form")`
`=e^(underset(xto0)lim((p_(1)a_(1)^(x)+p_(2)a_(2)^(x)+...+p_(n)a_(n)^(x))/(x)))`
`=e^(underset(xto0)lim(p_(1)a_(1)^(x)" ln "a_(1)+p_(2)a_(2)^(x)" ln "a_(2)+...+p_(n)a_(n)^(x)" ln "a_(n)))`
`=e^((p_(1)" ln "a_(1)+p_(2)" ln "a_(2)+...+p_(n)" ln "a_(n)))`
`=e^(("ln "a_(1)^(p_(1))+"ln "a_(2)^(p^(2))+...+"ln "a_(n)^(p^(n))))`
`=e^(("ln "a_(1)^(p_(1))a_(2)^(p^(2))+...a_(n)^(p^(n))))=a_(1)^(p^(1)).a_(2)^(p^(2))a_(3)^(p^(3))...a_(n)^(p^(n))`