Final answer:
The limit of (cosx - cosa)/(x - a) as x approaches a is equivalent to the derivative of the cosine function at a, which is evaluated to be -sin(a).
Step-by-step explanation:
The limit in question is a form of the derivative of the cosine function at point a. To find the limit of (cosx - cosa)/(x - a) as x approaches a, we can apply L'Hôpital's Rule, which allows us to differentiate the numerator and denominator separately and then take the limit if the limit originally yields an indeterminate form such as 0/0.
After differentiation, the numerator becomes -sin(x) and the denominator becomes 1. Taking the limit as x approaches a, we substitute x with a to find the limit:
Limit of (-sin(x))/(x - a) as x → a is -sin(a).
The limit of (cosx - cosa)/(x - a) as x approaches a is equivalent to the derivative of the cosine function at a, which is evaluated to be -sin(a).