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lim. xtan2x-2xtanx x->0 (1-cos2x)2 A)2B)-2C)1/2D)-1/2
1-cos2x=2sin^2xnow, the above lim becomesL= lim xtan2x-2xtanx * x^4 4x^4 sin^4xbut lim x^4/sin^4x as x tends to zero is 1now, use series expansionlim (2x^2+8x^4/3+O(x^5) – 2x^2 – 2x^4/3 – O(x^5)/(x^4)=8/3 – 2/3=2
1-cos2x=2sin^2xnow, the above lim becomesL= lim xtan2x-2xtanx * x^4 4x^4 sin^4xbut lim x^4/sin^4x as x tends to zero is 1now, use series expansionlim (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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