Let `a_1gt a_2gt a_3 ...a_gt1` `p_1gt p_2 gt ...p_n gt0` ; such that `p_1 + p_2+ p_3 + p_n= 1` . Also `F(x)=(p_1a_1^x+p_2a_2^x+...+p_nan^x)^(1/x)` Then lim x→∞F(x) equals

Options

A. `ln a_1`
B. `ln a_n`
C. `a_1`
D. `a_n`

Answer

D

Explanation

`Let underset(xtooo)limF(x)=L`
`:." "lnL=underset(xtooo)lim(p_(1)a_(1)^(x)lna_(1)+p_(2)a_(2)^(x)lna_(2)+...+p_(n)a_(n)^(x)lna_(n))/(p_(1)a_(1)^(x)+p_(2)a_(2)^(x)+...+p_(n)a_(n)^(x))`
Dividing by `(a_(n))^(x)` and taking `underset(xtooo)lim((a_(1))/(a_(n)))^(x),((a_(2))/(a_(n)))^(x)`,etc. vanish.
Therefore,
`lnL=(p_(n)lna_(n))/(p_(n))`
or `L=a_(n)`