Mathematics

Problem

A piecewise-defined function

p

is given by

p(x)=\begin{cases} \dfrac{\sqrt[3]{x+1}-1}{x}, & x>0, \\ \dfrac{\sqrt{x+1}-1}{x}, & -1<x<0. \end{cases}

Answer the following questions about the behavior of

p

x=0

. (a) Determine

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

. (b) Determine

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

. (c) Decide whether

p

can be extended to a continuous function at

x=0

. If so, state the necessary value of

p(0)

and justify your conclusion. (d) Discuss how the distinct radicals in the two pieces of

p

influence the existence (or non-existence) of the overall limit at

x=0