The value of `lim_(xrarr0)(1^x+2^x+3^x+...+n^x)^(a//x)/n`, is:

Options

A. `f(1^(+))+f(1^(-))=0`
B. `f(1^(+))+f(1^(-))+f(1)=3//2`
C. `f(-1^(+))+f(-1^(-))=-1`
D. `f(1^(+))+f(-1^(-))=0`

Answer

B

Explanation

`underset(xto0)lim((1^(x)+2^(x)+...+n^(x))/(n))^(1//x)=e^(underset(xtoo)lim((1^(x)-1)/(n)+(2^(x)-1)/(n)+...+(n^(x)-1)/(n))(1)/(x))`
`=e^(underset(xtoo)lim(1)/(n){(1^(x)-1)/(x)+(2^(x)-1)/(x)+...+(n^(x)-1)/(x)}`
`=e^((1)/(n)[log1+log2+...+logn])`
`=e^((1)/(n)(logn!))`
`=e^(log(n!)^((1)/(n)))=(n!)^((1)/(n))`