limx→01−cos3xxsin2x
By L'Hospital's rule
limx→0f(x)g(x)=limx→0f′(x)g′(x)
⇒ limx→00−3cos2x(−sinx)2xcos2x+sin2x [ Taking derivative of numerator and denominator ]
⇒ limx→03cos2xsinx2x(cos2x−sin2x)+2sinxcosx
⇒ 32limx→0cos2xsinxx(cos2x−sin2x)+2sinxcosx
⇒ 32limx→0sinxx(1−tan2x)+2tanx [ Dividing numerator and denominator by cos2x ]
⇒ 32limx→0cosxx(−2tanxsec2x)+(1−tan2x).1+sec2x [ Taking derivative of numerator and denominator ]
⇒ 32×11+1
⇒ 32×12
⇒ 34