To determine

To evaluate : limx→0x−sinxx−tanx 

Answer

limx→0x−sinxx−tanx =−12

Explanation

Calculations:

limx→0x−sinxx−tanx (00 Form)Applying l'Hospital's Rule=limx→01−cosx1−sec2x(00 Form)Applying l'Hospital's Rule again=limx→0sinx−2sec2xtanx=limx→0sinx−21cos2xsinxcosx=−12limx→0cos3x1=−12cos3(0)=−12

Conclusion: The value of the limit is   limx→0x−sinxx−tanx =−12