Define
h(x)=∣x−2∣+(x−2)2.
The graph of h appears smooth except possibly at x=2.
(a) Compute h(2+h)−h(2) for a generic non-zero h and simplify your result as far as possible.
(b) Evaluate the limits of hh(2+h)−h(2) as h→0− and h→0+. Show all intermediate steps, including absolute-value simplifications.
(c) Conclude whether h is differentiable at x=2. Support your conclusion with explicit reference to the limits obtained in part (b).
(d) Identify all real numbers x where h fails to be differentiable, and give a brief justification for each such point.
a. Part (a) — Algebraic preparation of the difference quotient
b. Part (b) — One-sided limit evaluations
c. Part (c) — Decision on differentiability at x=2
d. Part (d) — Full differentiability profile of h