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lim(xrarr0) (3 tan3x-4 tan2x-tanx)/(4x^(...

`lim_(xrarr0) (3 tan3x-4 tan2x-tanx)/(4x^(2)tanx)`

A

0

B

1

C

3

D

4

Text Solution

Verified by Experts

The correct Answer is:
D
`underset(xrarr0)(lim)(3tan3x-4tan2x-tanx)/(4x^(2)tanx)`
`=underset(xrarr0)(lim)(3[tan3x-tan2x]-(tan2x+tanx))/(4x^(2)tanx)`

`=underset(xrarr0)(lim)((3sinx)/(cos2x cos3x)-(sin3x)/(cosx cos2x))/(4x^(2)tanx)`
`=underset(xrarr0)(lim)((3)/(cos2xcos3x)-(3-4sin^(2)x)/(cosx cos2x))/((4x^(2))/(cosx))`
`=underset(xrarr0)(lim)(1)/(cos2xcos3x)(3cosx-(3-4sin^(2)x)cos3x)/(4x^(2))`
`=underset(xrarr0)(lim)(3(cosx-cos3x)+4sin^(2)x)/(4x^(2))`
`=underset(xrarr0)(lim)(6sinxsin2x+4sin^(2)x)/(4x^(2))`
`=underset(xrarr0)(lim)((3)/(2)(sinx)/(x)(2sin2x)/(2)+(sin^(2)x)/(x^(2)))`
`=3+1=4`
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