Chapter 13.2, Problem 36E

a.

To determine

To find: the values of limits.

Answer to Problem 36E

The values are $\underset{x\to 0^{+}}{\lim }h\left(x\right)=0,\underset{x\to 2^{-}}{\lim }h\left(x\right)=4,\underset{x\to 0}{\lim }h\left(x\right)=0,\underset{x\to 2^{+}}{\lim }h\left(x\right)=6,\underset{x\to 1}{\lim }h\left(x\right)=1,\underset{x\to 2}{\lim }h\left(x\right)=\text{does not exist}$ .

Explanation of Solution

Given information:

The function is $h\left(x\right)=\left\{\begin{array}{l}x\text{ if }x<0\\ x^2\text{ if }0<x\le 2\\ 8-x\text{ if }x>2\end{array}\right\}$ .

Calculation: to find the values of limit, substitute and simplify,

  $\begin{array}{l}\underset{x\to 0^{+}}{\lim }h\left(x\right)=\underset{x\to 0^{+}}{\lim }{x}^2\\ \underset{x\to 0^{+}}{\lim }h\left(x\right)=0\text{ [substitute and simplify]}\end{array}$

  $\begin{array}{l}\underset{x\to 2^{-}}{\lim }h\left(x\right)=\underset{x\to 2^{-}}{\lim }{x}^2\\ \underset{x\to 2^{-}}{\lim }h\left(x\right)=4\text{ [substitute and simplify]}\end{array}$

  $\begin{array}{l}\underset{x\to 0}{\lim }h\left(x\right)=\underset{x\to 0^{+}}{\lim }{x}^2\\ \underset{x\to 0}{\lim }h\left(x\right)=0\text{ [substitute and simplify]}\end{array}$

  $\begin{array}{l}\underset{x\to 2^{+}}{\lim }h\left(x\right)=\underset{x\to 2^{+}}{\lim }8-x\\ \underset{x\to 2^{+}}{\lim }h\left(x\right)=6\text{ [substitute and simplify]}\end{array}$

  $\begin{array}{l}\underset{x\to 1}{\lim }h\left(x\right)=\underset{x\to 1}{\lim }{x}^2\\ \underset{x\to 1}{\lim }h\left(x\right)=1\text{ [substitute and simplify]}\end{array}$

  $\underset{x\to 2}{\lim }h\left(x\right)=\text{does not exist}$

b.

To determine

To draw: the graph of function.

Explanation of Solution

Given information:

The function is $h\left(x\right)=\left\{\begin{array}{l}x\text{ if }x<0\\ x^2\text{ if }0<x\le 2\\ 8-x\text{ if }x>2\end{array}\right\}$ .

Graph: to draw the graph, use the plotting point method,

$x$	$h\left(x\right)$
$0$	$0$
$1$	$1$
$3$	$5$

Now, by using the above table, and join the points

The graph can be obtained as:

  

Interpretation: from the above graph, it can be observed that the $\underset{x\to 2}{\lim }f\left(x\right)$ does not exist.