Calculus: Early Transcendentals, Chapter 4, 4.4, Section 4.4, Problem 40

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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