Question

If f(x)=arcsin x then lim x→1/2 f(x) - f(1/2) / x-1/2:

a) 0
b) 1
c) [infinity]
d) π/2

Answer 1

The limit of the given expression is 0. The correct option is a.

Step-by-step explanation:

To find the limit of the expression, we can use the fact that the limit of the difference quotient is equal to the derivative of the function at that point. First, we find the derivative of f(x) = arcsin(x), which is f'(x) = 1/√(1 - x²). Then, we substitute the given value x = 1/2 into the derivative to find f'(1/2) = 1/√(1 - (1/2)²) = 1/√(3/4) = 2/√(3).

Next, we substitute f'(1/2) and x = 1/2 into the difference quotient: (f(x) - f(1/2))/(x - 1/2) = (arcsin(x) - arcsin(1/2))/(x - 1/2). Plugging in x = 1/2 gives (arcsin(1/2) - arcsin(1/2))/(1/2 - 1/2) = 0/0, which is an indeterminate form.

However, we can simplify the expression by using the fact that arcsin(1/2) = pi/6. Therefore, the limit becomes:

limx→1/2 (arcsin(x) - arcsin(1/2))/(x - 1/2) = limx→1/2 (pi/6 - pi/6)/(x - 1/2) = limx→1/2 0/(x - 1/2) = 0.

Hence, a is the correct option.