A piecewise function is given by h(x)= r(x) s(x)

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

Problem

A piecewise function

h

is given by

h(x)=\frac{r(x)}{s(x)},\qquad r(x)=\begin{cases}\tan(3x),&x<0,\\ x^3,&x\ge 0,\end{cases}\qquad s(x)=\begin{cases}x,&x<0,\\ \sin(2x),&x\ge 0.\end{cases}

Determine the following limits and justify your answers. (a) Evaluate

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

. Show all supporting work. (b) Evaluate

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

. Show all supporting work. (c) Based on your results in parts (a) and (b), state whether

\displaystyle\lim_{x\to 0}h(x)

exists and, if so, give its value.