Evaluate the following limit: `(lim)_(x->0)(xtan2x-2xtanx)/((1-cos2x)^2)`

Explanation

`underset(xto0)lim(xtan2x-2xtanx)/(4sin^(4)x)=underset(xto0)lim(x)/(4sin^(4)x)[(2tanx)/(1-tan^(2)x)-2tanx]`
`=underset(xto0)lim(xtan^(3)x)/(2sin^(4)x(1-tan^(2)x))`
`=1/2underset(xto0)lim(x)/(sinx).(1)/(cos^(3)x).(1)/(1-tan^(2)x)`
`=(1)/(2).1.(1)/(1^(3)
).(1)/(1-0)=1/2`