MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
Define the function h by
h(x)=⎩⎨⎧kxsinx,k,x=0,x=0,
where k is a real constant.
(1) Part 1: (a) Determine the value of k that guarantees continuity of h at x=0. Justify your conclusion.
(2) Part 2: (b) For the value of k found in part (a), determine whether h is differentiable at x=0. If it is differentiable, compute h′(0); if not, explain why differentiability fails.
(3) Part 3: (c) Discuss how the relationship between the limit in part (a) and the derivative in part (b) illustrates the connection between continuity and differentiability at a point.