Mathematics

Problem

A function

f

is defined by

f(x)=\begin{cases} \dfrac{x^2-9}{x-3}, & x\neq 3,\\[4pt] 2, & x=3. \end{cases}

Answer the following questions related to the behavior of

f

x=3

. (a) Identify the type of discontinuity, if any, that

f

exhibits at

x=3

. Explain your reasoning by comparing

\displaystyle\lim_{x\to3}f(x)

with

f(3)

. (b) Calculate

\displaystyle\lim_{x\to3}f(x)

analytically. (c) Redefine

f(3)

so that the resulting function is continuous at

x=3

. Justify why your definition removes the discontinuity. (d) Describe how the graph of

f

behaves in a neighborhood of

x=3

, including any relevant coordinate values.