Chapter 2.1, Problem 64E

(a)

To determine

To graph: The piecewise function $f\left(x\right)=\{\begin{array}{l}x,-1\le x<0,\text{or}0<x\le 1\\ 1,x=0\\ 0,x<-1,\text{or}x>1\end{array}$ .

Explanation of Solution

Given information: The function is $f\left(x\right)=\{\begin{array}{l}x,-1\le x<0,\text{or}0<x\le 1\\ 1,x=0\\ 0,x<-1,\text{or}x>1\end{array}$ .

Graph:

To graph $f\left(x\right)=\{\begin{array}{l}x,-1\le x<0,\text{or}0<x\le 1\\ 1,x=0\\ 0,x<-1,\text{or}x>1\end{array}$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator and press $\boxed{\text{Y}=}$ key. To write the function $f\left(x\right)=x$ for the interval $-1\le x<0,\text{or}0<x\le 1$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes as given below:

  $\begin{array}{l}\boxed{\text{X}}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{(-)}\boxed{1}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{\Rightarrow }\boxed{1}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{5}\boxed{0}\boxed{)}\\ \boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{\Rightarrow }\boxed{2}\boxed{(}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{3}\boxed{0}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{\Rightarrow }\boxed{1}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{6}\boxed{1}\boxed{)}\end{array}$

Press Enter. To write the function $f\left(x\right)=1$ for the interval $x=0$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes as given below:

  $\boxed{1}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{1}\boxed{0}\boxed{)}$

Again press Enter. To write the function $f\left(x\right)=0$ for $x<-1,\text{or}x>1$ after the symbol ${\text{Y}}_3$ . Enter the keystrokes as given below:

  $\boxed{0}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{5}\boxed{(-)}\boxed{1}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{\Rightarrow }\boxed{2}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{3}\boxed{1}\boxed{)}$

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}=\text{X}/\left(\text{X}\ge -1\text{and X}<0\right)\text{or}\left(\text{X}>0\text{and X}\le 1\right)\\ {\text{Y}}_2=1/\left(\text{X}=0\right)\\ {\text{Y}}_3=0/\left(\text{X}<-1\text{or}\text{X}>1\right)\end{array}$

Now, press the $\boxed{\text{GRAPH}}$ key to produce the graph of given function in standard window as shown below. Set the window at $\left[-4.7,4.7\right]$ by $\left[-3.1,3.1\right]$ .

  

Figure (1)

Interpretation: The graph of the piecewise function shows is a horizontal line except the interval $\left[-1,1\right]$ and equation of line $y=x$ .

(b)

To determine

To find: The points $c$ in the domain of $f\left(x\right)$ such that $\underset{x\to c}{\lim }f\left(x\right)$ exist for the given function.

Answer to Problem 64E

The limit $\underset{x\to c}{\lim }f\left(x\right)$ exist for every $c$ in the interval $\left(-\infty ,-1\right)\cup \left[-1,1\right]\cup \left(1,\infty \right)$ .

Explanation of Solution

Given information: The function is $f\left(x\right)=\{\begin{array}{l}x,-1\le x<0,\text{or}0<x\le 1\\ 1,x=0\\ 0,x<-1,\text{or}x>1\end{array}$ .

Calculation:

The function $f\left(x\right)=x$ corresponds for all the values of $x$ in $\left(-1,1\right)$ . So, the limit $\underset{x\to c}{\lim }f\left(x\right)$ exists for every $x$ in the interval $\left[-1,1\right]$ .

The function exist for all points less than $-1$ and greater than 1. So, the limit $\underset{x\to c}{\lim }f\left(x\right)$ exists for every $x$ in the interval $\left(-\infty ,-1\right]\cup \left[1,\infty \right)$ .

Therefore, the limit $\underset{x\to c}{\lim }f\left(x\right)$ exist for every $c$ in the interval $\left(-\infty ,-1\right)\cup \left[-1,1\right]\cup \left(1,\infty \right)$ .

(c)

To determine

To find: The points $c$ in the domain of $f\left(x\right)$ such that only left hand limit exist at $x=c$ .

Answer to Problem 64E

There is no point $c$ such that only the left hand limit exists.

Explanation of Solution

Given information: The function is $f\left(x\right)=\{\begin{array}{l}x,-1\le x<0,\text{or}0<x\le 1\\ 1,x=0\\ 0,x<-1,\text{or}x>1\end{array}$ .

Calculation:

As observed from the graph in part (a), the function is defined for every point in the domain of the function. So, left hand limit exist at every point.

Therefore, there is no point $c$ such that only the left hand limit exists.

(d)

To determine

To find: The points $c$ in the domain of $f\left(x\right)$ such that only right hand limit exist at $x=c$ .

Answer to Problem 64E

There is no point $c$ such that only the left hand limit exists.

Explanation of Solution

Given information: The function is $f\left(x\right)=\{\begin{array}{l}x,-1\le x<0,\text{or}0<x\le 1\\ 1,x=0\\ 0,x<-1,\text{or}x>1\end{array}$ .

Calculation:

As observed from the graph in part (a), the function is defined for every point in the domain of the function. So, right hand limit exist at every point.

Therefore, there is no point $c$ such that only the right hand limit exists.