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lim. xtan2x-2xtanx x->0 (1-cos2x) 2 A)2 B)-2 C)1/2 D)-1/2

lim.    xtan2x-2xtanx  
 x->0   (1-cos2x)2
 
A)2
B)-2
C)1/2
D)-1/2
    

Grade:

2 Answers

Aditya Gupta
2080 Points
3 years ago
1-cos2x=2sin^2x
now, the above lim becomes
L= lim xtan2x-2xtanx * x^4
        4x^4                     sin^4x
but lim x^4/sin^4x as x tends to zero is 1
now, use series expansion
lim (2x^2+8x^4/3+O(x^5) – 2x^2 – 2x^4/3 – O(x^5)/(x^4)
=8/3 – 2/3=2
Aditya Gupta
2080 Points
3 years ago
 
1-cos2x=2sin^2x
now, the above lim becomes
L= lim xtan2x-2xtanx x^4
        4x^4                     sin^4x
but lim x^4/sin^4x as x tends to zero is 1
now, use series expansion
lim (2x^2+8x^4/3+O(x^5) – 2x^2 – 2x^4/3 – O(x^5)/(4x^4)
=(8/3 – 2/3)/4
=1/2

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