If `f:[-1,1]toR` and `f'(0)=lim_(ntooo)nf(1/n)` and `f(0)=0`. Find the value of `lim_(ntooo)2/(pi)(n+1)cos^(-1)(1/n)-n` given that `0lt|lim_(to oo)cos^(-1)(1/n)|lt (pi)/2`.

Options

A. `(2-pi)/(pi)`
B. `(pi-2)/(pi)`
C. `1`
D. `0`

Answer

B

Explanation

`lim_(n to oo) {(n+1)(2)/(pi)cos^-1.(1)/(n)-n}`
`=lim_(n to oo) (2)/(pi){(n+1)cos^-1.(1)/(n)-(pi)/(2)n}`
`=lim_(n to oo) (2)/(pi){n(cos^-1.(1)/(n)-(pi)/(2))+cos^-1.(1)/(n)}`
`=lim_(n to oo) (2)/(pi){nsin^-1.(1)/(n)-(pi)/(2)+cos^-1.(1)/(n)}`
`=lim_(n to oo) (2)/(pi){(sin^-1.(1)/(n))/((1)/(n))+cos^-1.(1)/(n)}=(2)/(pi)(-1+(pi)/(2))=(pi-2)/(pi)`