`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.

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Textbook Question

Chapter 4, 4.4 - Problem 40 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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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.`

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