lim : xtan2x-2xtanx

(x tends to 0) (1-cos2x)²

^{equals to}

^1/4

¹

^1/2

^-1/2

Take x common and 1-cos2x=2(sin^2 x). hence x [tan2x - 2tanx]/{2(sin^2 x)}^2 is the equation)

tan2x = 2tanx/(1 - tan^2 x). On substituting and simplifying we get x[2tanx . tan^2 x] / (1 - tan^2 x)(4 sin^4 x)

Write tanx = sinx/cosx. On substituting and simplifying we get 2x / 4sinxcosx (cos^2 x - sin^2 x)

Write cos^2 x = 1 - sin^2 x and take numerator x to denominator, we get 2 / [4 (sinx/x)cosx - 8 (sinx/x)cosx sin^2 x]

Lim x -> 0 sinx/x = 1, therefore 2/(4-0) = 2/4 = 1/2