The value of `lim_(x to oo)(2x^(1//2)+3x^(1//3)+4x^(1//4)+…………+nx^(1//n))/((2x-3)^(1//2)+(2x-3)^(1//3)+……….+(2x-3)^(1//n))` is

Options

A. 0
B. 2
C. `sqrt(2)`
D. `(1)/(sqrt(3))`

Answer

C

Explanation

Let
`l=lim_(xto oo)(2x^(1//2)+3x^(1//3)+4x^(1//4)+....+nx^(1//n))/((2x-3)^(1//2)+(2x-3)^(1//3)+....+(2x-3)^(1//n))`
Then, ,
`l=lim_(xto oo)(2h^(-1//2)+3h^(-1//3)+4h^(-1//4)+....+nx^(-1//n))/((2-3h)^(1//2)+(h)^(-1//2)+(2-3h)^(1//3)+h^(-1//3)....+(2-3h)^(1//n)h^(-1//n)) "where" h=(1)/(x)`
`rArrl=lim_(hto0)(2+3h^(((1)/(2)-(1)/(3)))+4h^(((1)/(2)-(1)/(4)))+.....+nh^(((1)/(2)-(1)/(n))))/((2-3h)^(1/2)+(2-3h)^(1/3)+h^(((1)/(2)-(1)/(4)))+....+(2-3h)^(1/n)h^(((1)/(2)-1/n)))`
`rArr l=(2)/(sqrt(2))=sqrt(2)`