Limits & Continuity: lim_xto0fracxtan2x-2xtan x(1-cos2x)^2 equals. | JEE Main 2018 | Yukti Vault

Yukti Vault

JEE Main 2018 (Online) 15th April Evening ShiftQuestion 1

$\lim_{x\to0}\frac{x\tan2x-2x\tan x}{(1-\cos2x)^{2}}$ equals.

Limits & Continuity

$1$$-\frac12$$\frac14$$\frac12$✓ Correct answer

Solution

$x\tan2x-2x\tan x=x\left(2x+\frac{8x^3}3-2x-\frac{2x^3}3\right)+\dots=2x^4+\dots$ and $(1-\cos2x)^2=4\sin^4x\approx4x^4$, so the limit is $\frac12$.

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