The value of `lim_(ntooo) [(1+2+3+...+n)/n^2]` is

Answer

B

Explanation

`cos(tanx)-cosx=2sin((x+tanx)/(2))sin((x-tanx)/(2))`
or `underset(xto0)lim(cos(tanx)-cosx)/(x^(4))`
`=underset(xto0)lim(2sin((x+tanx)/(2))sin((x-tanx)/(2)))/(x^(4))`
`=underset(xto0)lim(2sin((x+tanx)/(2))sin((x-tanx)/(2)))/(x^(4)((x+tanx)/(2))((x-tanx)/(2)))((x^(2)-tan^(2)x)/(4))`
`=(1)/(2)underset(xto0)lim(x^(2)-tan^(2)x)/(x^(4))`
`=(1)/(2)underset(xto0)lim(x^(2)-(x+(x^(3))/(3)+(2)/(15)x^(5)+...)^(2))/(x^(4))`
`=(1)/(2)underset(xto0)lim(1)/(x^(2))(1-(1+(x^(2))/(3)+(2)/(15)x^(4)+...)^(2))=-(1)/(3)`