Final answer:
The limit lim f(x)−f(12)/x−12 as x approaches 12 is c) undefined.
Step-by-step explanation:
To find the value of the given limit, let's first find f(12). Since f(x) = arcsin(x), we substitute 12 into the function:
f(12) = arcsin(12).
However, the range of the arcsin function is restricted to -π/2 to π/2. Therefore, f(12) is undefined.
Next, let's find the limit:
lim f(x)−f(12)/x−12 as x approaches 12.
We substitute x = 12 into the limit expression:
lim f(x)−f(12)/x−12 = lim f(12)−f(12)/12−12 = lim 0/0.
Since we have a 0/0 indeterminate form, we can use L'Hopital's Rule.
Differentiating f(x) with respect to x, we get f'(x) = 1/√(1-x^2).
Now we can apply L'Hopital's Rule again:
lim f(x)−f(12)/x−12 = lim (1/√(1-x^2))/(1-0) = lim (1/√(1-x^2)).
Substituting x = 12, we get
lim (1/√(1-(12)^2)) = lim (1/√(1-144)) = lim (1/√(-143)).
Since the square root of a negative number is undefined, the limit is also undefined.