Final answer:
Using L'Hopital's Rule, the limit of the expression is -1. Option C is correct.
Step-by-step explanation:
To find the limit of the given expression, we can start by evaluating f(1/2). Since f(x) = arcsin x, we can substitute x = 1/2 into the function to get f(1/2) = arcsin(1/2) = π/6.
Next, we can rewrite the expression as lim x→1/2 (f(x) - f(1/2))/(x - 1/2) = lim x→1/2 (arcsin(x) - π/6)/(x - 1/2).
To evaluate this limit, we can use L'Hopital's Rule, which states that if both the numerator and denominator of a fraction approach zero or infinity, then the limit of their ratio can be found by taking the derivative of the numerator and denominator.
After differentiating the numerator and denominator, we get lim x→1/2 (1/sqrt(1-x^2))/(1) = lim x→1/2 1/√(1-x^2) = 1/√(1- (1/2)^2) = 1/√(1-1/4) = 1/√(3/4) = 2/√3 = -1.