MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
The function f is defined by f(x)=∣x∣. Which statement best explains why f is not differentiable at x=0?
(a) Both the left–hand and right–hand difference quotients approach +∞, so the derivative is undefined at x=0.
(b) The function is not continuous at x=0, so a derivative cannot exist there.
(c) The limit h→0limhf(0+h)−f(0) does not exist because the left–hand and right–hand limits are different.
(d) Because the slopes of the two line segments are −1 and 1, the derivative at x=0 is their average, 0, giving a horizontal tangent.