Final answer:
To find the limit of the given expression, rewrite it in terms of u and v, then use the limit rule for sin(x)/x. The limit is cos(v) / (2 * √a).
Step-by-step explanation:
To find the limit of the given expression, we can use the fact that the limit of sin(x)/x as x approaches 0 is 1. We can rewrite the expression as:
lim (x → a) [sin(√x) - sin(√a)] / (x - a)
= lim (x → a) [(sin(√x) - sin(√a))/(√x - √a)] * [(√x - √a)/(x - a)]
By substituting u = √x and v = √a, we can rewrite the expression as:
= lim (u → v) [sin(u) - sin(v)] / (u² - v²)
= lim (u → v) [sin(u) - sin(v)] / [(u - v)(u + v)]
= lim (u → v) [(sin(u) - sin(v))/(u - v)] * 1/(u + v)
= cos(v) / (2 * √a)