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Lim x→0(1-cos 2x)/(xtan x) equals Single correct option
lim x → 0 1 − cos 2 x x tan x \displaystyle\lim_{x\to0}\frac{1-\cos2x}{x\tan x} x → 0 lim x tan x 1 − cos 2 x equals
A. 1 1 1 B. 4 4 4 C. 1 2 \frac12 2 1 D. 2 2 2 Show answer and solution Answer: (D) 2 2 2
1 − cos 2 x = 2 sin 2 x 1-\cos2x=2\sin^2x 1 − cos 2 x = 2 sin 2 x , so the expression is
2 sin 2 x x tan x = 2 ⋅ sin x x ⋅ sin x tan x → 2 ⋅ 1 ⋅ 1 = 2 \frac{2\sin^2x}{x\tan x}=2\cdot\frac{\sin x}{x}\cdot\frac{\sin x}{\tan x}\to2\cdot1\cdot1=2 x t a n x 2 s i n 2 x = 2 ⋅ x s i n x ⋅ t a n x s i n x → 2 ⋅ 1 ⋅ 1 = 2 . Using
1 − cos 2 x ≈ 2 x 2 1-\cos 2x\approx 2x^2 1 − cos 2 x ≈ 2 x 2 and
x tan x ≈ x 2 x\tan x\approx x^2 x tan x ≈ x 2 agrees. The value
4 4 4 forgets that
( 2 x ) 2 2 \frac{(2x)^2}{2} 2 ( 2 x ) 2 gets divided by
x 2 x^2 x 2 only once.
Original question written in the NDA 2017 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake