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lim x→0 [x tan 2x − 2x tan x]/(1 − cos 2x)² equals -

1. 1
2. −1/2
3. 1/4
4. 1/2

Correct answer: 1/2

Solution

(1-cos2x)^2 = 4 sin^4 x ~ 4x^4. Numerator x(tan2x - 2tanx) = x*2t^3/(1-t^2) ~ 2x^4 with t=tanx. So the limit = 2x^4 / 4x^4 = 1/2.

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