Mathematics

Problem

Let the piecewise function

f(x)=\begin{cases} 2x-1, & x<3,\\ x^{2}-8, & x\ge 3 \end{cases}

be defined for all real

x

. a. Part 1: Determine

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

and

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

. Show your work. b. Part 2: Decide whether

\displaystyle \lim_{x\to 3} f(x)

exists. Explain your reasoning. c. Part 3: Evaluate

f(3)

and, using your results from parts (a) and (b), determine whether

f

is continuous at

x=3

. Provide a clear justification tied to the definition of continuity.