Let g(x)= cases |x| x , & x 0, 2, & x=0

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Let

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

Analyze the behavior of

g

x=0

. (1) Part 1: (a) Rewrite

g

for

x\ne 0

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

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

and

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

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

x=0

. Compute the corresponding jump size. (4) 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.