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Evaluate lim(xto0) (xtan2x-2xtanx)/((1-c...

Evaluate `lim_(xto0) (xtan2x-2xtanx)/((1-cos2x)^(2)).`

Text Solution

Verified by Experts

The correct Answer is:
`1//2`
`underset(xto0)lim(xtan2x-2xtanx)/(4sin^(4)x)=underset(xto0)lim(x)/(4sin^(4)x)[(2tanx)/(1-tan^(2)x)-2tanx]`
`=underset(xto0)lim(xtan^(3)x)/(2sin^(4)x(1-tan^(2)x))`
`=1/2underset(xto0)lim(x)/(sinx).(1)/(cos^(3)x).(1)/(1-tan^(2)x)`
`=(1)/(2).1.(1)/(1^(3) ).(1)/(1-0)=1/2`
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