Mathematics

Problem

Let the function

g

be defined by

g(x) = \begin{cases} |x-3| + 1, & x \neq 3,\\ 2, & x = 3. \end{cases}

Answer the following questions about

g

. a. Part 1: (a) Rewrite

|x-3|

as an explicit piecewise expression valid for all real

x

. Clearly indicate the branch that applies on each side of

x=3

. b. Part 2: (b) Using your result from part (a), determine

\displaystyle \lim_{x\to 3^-} g(x)

and

\displaystyle \lim_{x\to 3^+} g(x)

. Show algebraic justification for each one-sided limit. c. Part 3: (c) Decide whether

g

is continuous at

x=3

. Provide a concise argument referencing the relationship among the one-sided limits, the two-sided limit, and the value

g(3)

. Explain your reasoning in complete sentences.