For the function f(x)=∣x∣, which of the following best explains why f′(0) does NOT exist?
a) The slopes of the tangent lines on both sides approach ∞, creating a vertical tangent instead of a finite slope.
b) The function is not continuous at x=0, so a derivative cannot be taken there.
c) The limits of the difference quotient from the left and right approach different finite values, so the one–sided derivatives are unequal.
d) Using f′(x)=∣x∣x and substituting x=0 gives 00, an undefined value, so the derivative does not exist.