Mathematics

Problem

Let

g(x)=\begin{cases} \dfrac{|x|}{x}, & x\ne 0,\\ 2, & x=0. \end{cases}

Analyze the behavior of

g

x=0

. a. Part 1: (a) Rewrite

g

for

x\ne 0

as an explicit piecewise-defined expression that contains no absolute value symbols. b. Part 2: (b) Evaluate

\displaystyle \lim_{x\to 0^-}g(x)

and

\displaystyle \lim_{x\to 0^+}g(x)

. Show your work. c. Part 3: (c) Identify the type of discontinuity that occurs at

x=0

. Compute the corresponding jump size. d. Part 4: (d) State the function value that

g(0)

would need in order to remove the discontinuity at

x=0

, or explain why no such value exists.