The value of `lim_(xrarr(3pi)/(4)) (1+root3(tanx))/(1-2cos^(2)x)` is

Options

A. `-1//2`
B. `-2//3`
C. `-3//2`
D. `-1//3`

Answer

D

Explanation

`L=underset(xrarr3pi)(lim)(1+(tanx)^((1)/(3)))/(1-2 cos^(2)x)`
`=underset(xrarr3pi)(lim)(1+(tanx)^((1)/(3)))/(-cos2x)" "((0)/(0)" form")`
Applying L'Hospital's rule
`therefore" "L=underset(xrarr(3pi)/(4))(lim)((1)/(3)(tanx)^((-2)/(3))xxsec^(2)x)/(2sin2x)`
`=(-1)/(6)(-sec.(pi)/(4))^(2)`
`=(-1)/(6)xx2=-(1)/(3)`