`lim_(x->0) (x - sin(x))/(x - tan(x))` Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule...

`lim_(x->0) (x - sin(x))/(x - tan(x))` Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.

Expert Answers
tiburtius eNotes educator| Certified Educator

`lim_(x->0)(x-sin x)/(x-tan x)`

Apply L'Hospital's rule.

`=lim_(x->0)(1-cos x)/(1-1/cos^2x)`

Apply L'Hospital's rule again.

 `=lim_(x->0)sin x/(-(2sin x)/cos^3x)`

Cancel `sin x.`

`lim_(x->0)-(cos^3x)/2=-(cos^3 0)/2=-1/2`

` `The solution is `-1/2.`