Consider the function f(x)=3x. Which of the following best explains why f′ does not exist at x=0?
a) The left- and right-hand limits of the difference quotient are unequal, so a sharp corner occurs at x=0.
b) The tangent line at x=0 is vertical, so the slope is undefined.
c) Differentiating gives f′(x)=31x−2/3, and substituting x=0 yields f′(0)=0, so the slope is zero at x=0.
d) The left-hand derivative approaches −∞ while the right-hand derivative approaches +∞, so the one-sided limits are unequal.