# `lim_(x->a^+) (cos(x)ln(x - a))/(x - sin(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->a^+) (cos(x)ln(x - a))/(x - sin(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 40 - Calculus: Early Transcendentals (7th Edition, James Stewart).
See all solutions for this textbook.

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to evaluate the limit, hence, you need to replace `a^+` for x in limit, such that:

`lim_(x->a^+) (cos x*ln (x - a))/(x - sin x) = (cos a*ln (a - a))/(a- sin a)`

Notice that `ln(a - a) -> -oo,` hence `lim_(x->a^+) (cos x*ln (x - a))/(x - sin x) = ((cos a)/(a- sin a))(-oo) = -oo.`

It is no need to use l'Hospital's rule, since you did not obtained an indetermination. The limit can be directly evaluated.

Hence, evaluating the given limit, yields `lim_(x->a^+) (cos x*ln (x - a))/(x - sin x) = -oo.`

We’ve answered 319,630 questions. We can answer yours, too.