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underset(x to 0)(lim) (x tan 2X - X tan ...

`underset(x to 0)(lim) (x tan 2X - X tan x)/((1 - cos 2X)^(2))` equals

A

2

B

`-2`

C

`1/2`

D

`-1/2`

Text Solution

Verified by Experts

The correct Answer is:
C
`underset( x to 0) lim (x tan 2x - 2x tan x)/((1-cos 2x)^(2))`
NOTE In trigonometry try to make all trigonometric function in same angle. It is called 3rd Golden rule of trigonometry.
`underset( x to 0) lim (x(2 tan x)/(1- tan^(2) x) - 2x tan x)/((2 sin^(2) x)^(2))`
` underset( x to 0) lim(2x tan x[1/(1-tan^(2)x)-1])/(4 sin^(4) x) `
` underset( x to 0)lim (2x tan x[(1-1+tan^(2) x)/(1-tan^(2)x)])/(4 sin^(4) x) `
` underset(x to 0) lim (2x tan^(3) x)/(2 sin^(4) x(1-tan^(2) x))=underset( x to 0) lim 1/2 (x((tan x)/x)^(3))/(sin^(4) x (1-tan^(2) x))`
`1/2 underset(x to 0) lim (((tan x)/x)^(3))/(((sin x)/x)^(4) (1-tan^(2) x))=(1*(1)^(3))/(2(1)^(4)(1-0))=1/2`
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