Calculus limits question.

Question

If f is a function such that lim x->2 (f(x) - f(2))/(x-2) = 0, which of the following must be true?

(A) The limit of f(x) as x approaches 2 does not exist.

(B) f is not defined at x = 2;

(C) The derivative of f at x = 2 is 0.

(D) f is continuous at x = 0;

(E) f(2) = 0

I can't figure this one out, Thanks.

Answer 1

f'(a) = lim_x→a[(f(x) - f(a)) / (x - a)]

So, lim_x→2 [(f(x) - f(2)) / (x - 2] = f'(2) = 0

Answer: C