Mathematics

Problem

A function

h

is defined by

h(x) = \begin{cases} \ln(x+2), & -2 < x < 1,\\ a, & x = 1,\\ \sqrt{x-1} + 1, & x > 1. \end{cases}

Answer the following questions. a. Part 1: (a) Determine

\displaystyle \lim_{x\to 1^-} h(x)

and

\displaystyle \lim_{x\to 1^+} h(x)

. Show all steps, including domain considerations. b. Part 2: (b) Find the value of

a

that makes

h

continuous at

x = 1

. Provide justification that references your work from part (a). c. Part 3: (c) State all real numbers

c

for which

h

is not continuous. Give reasons that account for both the function’s domain and any discontinuities of removable, jump, or infinite type.