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lim : xtan2x-2xtanx (x tends to 0) (1-cos2x) 2 equals to 1/4 1 1/2 -1/2

lim                      :                xtan2x-2xtanx
(x tends to 0)                       (1-cos2x)2  
 
equals to 
1/4
1
1/2
-1/2
 

Grade:

1 Answers

Deepak Kumar Shringi
askIITians Faculty 4405 Points
2 years ago
Take x common and 1-cos2x=2(sin^2 x). hence x [tan2x - 2tanx]/{2(sin^2 x)}^2 is the equation)

tan2x = 2tanx/(1 - tan^2 x). On substituting and simplifying we get x[2tanx . tan^2 x] / (1 - tan^2 x)(4 sin^4 x)

Write tanx = sinx/cosx. On substituting and simplifying we get 2x / 4sinxcosx (cos^2 x - sin^2 x)

Write cos^2 x = 1 - sin^2 x and take numerator x to denominator, we get 2 / [4 (sinx/x)cosx - 8 (sinx/x)cosx sin^2 x]

Lim x -> 0 sinx/x = 1, therefore 2/(4-0) = 2/4 = 1/2

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