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Evaluate the limit lim_(x→ 0)(xtan(2x)-2xtan x)/((1-cos 2x)²).

1. 2
2. -2
3. 1/2
4. -1/2

Correct answer: 1/2

Solution

As x->0, x tan2x - 2x tan x = x[(2x+8x^3/3)-2(x+x^3/3)] = x(2x^3)=2x^4 to leading order, and (1-cos2x)^2=(2x^2)^2=4x^4. Limit = 2/4 = 1/2.

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