Chapter 2.3, Problem 43E

To determine

To graph: The function $f\left(x\right)$ such that $f\left(4\right)$ exists, $\underset{x\to 4}{\lim }f\left(x\right)$ exist but $f\left(x\right)$ is not continuous at $x=4$.

Explanation of Solution

Given information: The function $f\left(x\right)$ at $x=4$ exists, $\underset{x\to 4}{\lim }f\left(x\right)$ exist but $f\left(x\right)$ is not continuous at $x=4$.

Graph:

It is given that right hand limit at $x=4$ exist. So, the right hand and left hand are equal. It means that the function breaks at point $-2$.

Also, the function is not continuous at $x=4$ it means that there is break or jump on the graph at this point. There exist many possibilities to draw the graph with these conditions. Let consider a function with given possibilities as:

  $f\left(x\right)=\{\begin{array}{l}x-3,x>4\\ 2,x=4\\ 1,x<4\end{array}$

To graph a function $f\left(x\right)=\{\begin{array}{l}x-3,x>4\\ 2,x=4\\ 1,x<4\end{array}$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $f\left(x\right)=x-3$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{\text{X}}\boxed{-}\boxed{3}\boxed{)}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{3}\boxed{4}\boxed{)}$ . Now press the $\boxed{\text{ENTER}}$ button and enter right hand side of the equation $f\left(x\right)=2$ after the symbol ${\text{Y}}_3$ . Enter the keystrokes $\boxed{(}\boxed{2}\boxed{)}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{1}\boxed{4}\boxed{)}$ . Now, press the $\boxed{\text{ENTER}}$ button and enter right hand side of the equation $f\left(x\right)=1$ after the symbol ${\text{Y}}_3$ . Enter the keystrokes $\boxed{(}\boxed{1}\boxed{)}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{5}\boxed{4}\boxed{)}$.

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}=\left(\text{X}-3\right)/\left(\text{X}>4\right)\\ {\text{Y}}_2=\left(2\right)/\left(\text{X}=4\right)\\ {\text{Y}}_3=\left(1\right)/\left(\text{X}<4\right)\end{array}$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown below.

  

Figure (1)

As observed from the graph, the left hand limit of $f\left(x\right)$ at $x=4$ is $1$ and the right hand limit is $1$ . The left hand limit of $f\left(x\right)$ is equal to the right hand limit at $x=4$ . So, the value of $\underset{x\to 4}{\lim }f\left(x\right)$ exists and is equal to $1$.

The value of $f\left(x\right)$ at $x=4$ is equal to $2$ . So, the value of $f\left(4\right)$ exists.

Interpretation: The value $\underset{x\to 4}{\lim }f\left(x\right)$ is not equal to the value of $f\left(4\right)$ . So, the function $f\left(x\right)$ is not continuous at $x=4$ . So, one possibility for the function is $f\left(x\right)=\{\begin{array}{l}x-3,x>4\\ 2,x=4\\ 1,x<4\end{array}$.