Chapter 2.3, Problem 44E

To determine

To graph: The graph of the function $f\left(x\right)$ such that $f\left(x\right)$ is continuous for all the values of $x$ except $x=1$ , where $f\left(x\right)$ has a non removable discontinuity.

Explanation of Solution

Given information: The condition for the function is $f\left(x\right)$ is continuous for all the values of $x$ except $x=1$ , where $f\left(x\right)$ has a non removable discontinuity.

Graph:

To graph a function $f\left(x\right)=\frac{1}{x-1}$ , 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 $f\left(x\right)=\frac{1}{x-1}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{1}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{-}\boxed{1}\boxed{)}$.

The display will show the equation,

  ${\text{Y}}_{\text{1}}=1/\left(\text{X}-1\right)$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown in Figure (1).

  

Figure (1)

As observed from the graph, the left hand limit of $f\left(x\right)$ at $x=1$ is $-\infty$ and the right hand limit is $\infty$.

The left hand limit of $f\left(x\right)$ is not equal to the right hand limit at $x=1$ . So, the value of $\underset{x\to 1}{\lim }f\left(x\right)$ doe snot exists. So, the function is discontinuous at $x=1$.

Interpretation: The value of $\underset{x\to 1}{\lim }f\left(x\right)$ does not exists. So, the discontinuity at $x=1$ is non-removable. The function $f\left(x\right)$ is continuous for all the values of $x$ except $x=1$ . So, one possibility for the function is $f\left(x\right)=\frac{1}{x-1}$.