`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.

Asked on by enotes

Textbook Question

Chapter 4, 4.4 - Problem 28 - Calculus: Early Transcendentals (7th Edition, James Stewart).
See all solutions for this textbook.

1 Answer | Add Yours

tiburtius's profile pic

tiburtius | High School Teacher | (Level 2) Educator

Posted on

`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.`  

We’ve answered 318,982 questions. We can answer yours, too.

Ask a question