Chapter 2.2, Problem 6E

(a)

To determine

To find: The value of $\underset{x\to \infty }{\lim }f\left(x\right)$ for $f\left(x\right)=\frac{2x-1}{\left|x\right|-3}$ with the help of graph and table.

Answer to Problem 6E

The value of $\underset{x\to \infty }{\lim }f\left(x\right)$ for $f\left(x\right)=\frac{2x-1}{\left|x\right|-3}$ is $2$ .

Explanation of Solution

Given information:The function is $f\left(x\right)=\frac{2x-1}{\left|x\right|-3}$ .

Calculation:

Find the limit graphically.To graph a function $y=\frac{2x-1}{\left|x\right|-3}$ , 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 $y=\frac{2x-1}{\left|x\right|-3}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{2}\boxed{\text{X}}\boxed{-}\boxed{1}\boxed{)}\boxed{÷}\boxed{(}\boxed{\text{MATH}}\boxed{\to }\boxed{1}\boxed{\text{X}}\boxed{)}\boxed{-}\boxed{3}\boxed{)}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=\left(\text{2X}-1\right)/\left(\text{abs}\left(\text{X}\right)-3\right)$

Press the window key and adjust the window to $\left[-20,20\right]$ by $\left[-4,4\right]$ .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 below.

  

Figure (1)

As observed from graph, the function $f\left(x\right)=\frac{2x-1}{\left|x\right|-3}$ tends to $2$ as $x$ tends to $\infty$ .

Find the limit by table.To make the table of function $y=\frac{2x-1}{\left|x\right|-3}$ , 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 $y=\frac{2x-1}{\left|x\right|-3}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{2}\boxed{\text{X}}\boxed{-}\boxed{1}\boxed{)}\boxed{÷}\boxed{(}\boxed{\text{MATH}}\boxed{\to }\boxed{1}\boxed{\text{X}}\boxed{)}\boxed{-}\boxed{3}\boxed{)}$ .

First set the Table setup, enter the keystrokes $\boxed{2\text{nd}}\boxed{\text{WINDOW}}$ , then set to independent variable to ask.

Now draw the table, enter the keystrokes $\boxed{2\text{nd}}\boxed{\text{GRAPH}}$ , and then enter the values of $\text{X}$ to $100$ , $200$ , $300$ , $400$ , $-100$ , $-200$ and $-300$ . The table is shown in window.

  

Figure (2)

As observed from table the value of $\text{Y}$ approaches to $2$ as $x$ approaches to $\infty$ .

Therefore, the value of $\underset{x\to \infty }{\lim }f\left(x\right)$ for $f\left(x\right)=\frac{2x-1}{\left|x\right|-3}$ is $2$ .

(b)

To determine

To find:The value of $\underset{x\to -\infty }{\lim }f\left(x\right)$ for $f\left(x\right)=\frac{2x-1}{\left|x\right|-3}$ with the help of graph and table.

Answer to Problem 6E

The value of $\underset{x\to -\infty }{\lim }f\left(x\right)$ for $f\left(x\right)=\frac{2x-1}{\left|x\right|-3}$ is $-2$ .

Explanation of Solution

Given information:The function is $f\left(x\right)=\frac{2x-1}{\left|x\right|-3}$ .

Calculation:

To find the limit of the function $f\left(x\right)=\frac{2x-1}{\left|x\right|-3}$ with the help of the graph as $x$ tends to $-\infty$ , consider Figure (1) drawn in part (a). From the graph it can be observed that the function tends to $-2$ as $x$ tends to $-\infty$ .

To find the limit of the function $f\left(x\right)=\frac{2x-1}{\left|x\right|-3}$ with the help of the table as $x$ tends to $-\infty$ , consider Figure (2) drawn in part (a). From the table it can be observed that the function approaches to $-2$ as $x$ approaches to a big value on the left side of number line or to $-\infty$ .

Therefore, the value of $\underset{x\to -\infty }{\lim }f\left(x\right)$ for $f\left(x\right)=\frac{2x-1}{\left|x\right|-3}$ is $-2$ .

(c)

To determine

To find:The horizontal asymptote to the function $f\left(x\right)=\frac{2x-1}{\left|x\right|-3}$ .

Answer to Problem 6E

The horizontal asymptote to the function $f\left(x\right)=\frac{2x-1}{\left|x\right|-3}$ are $y=2$ and $y=-2$ .

Explanation of Solution

Given information:The function is $f\left(x\right)=\frac{2x-1}{\left|x\right|-3}$ .

Calculation:

The horizontal asymptote of the graph of a function $y=f\left(x\right)$ is the line $y=a$ if $\underset{x\to \infty }{\lim }f\left(x\right)=a$ or $\underset{x\to -\infty }{\lim }f\left(x\right)=a$ .

From part (a) and (b), it is known that for the function $f\left(x\right)=\frac{2x-1}{\left|x\right|-3}$ right hand limit is $\underset{x\to \infty }{\lim }f\left(x\right)=2$ and left hand limit is $\underset{x\to -\infty }{\lim }f\left(x\right)=-2$ . So, the horizontal asymptote to the curve $y=f\left(x\right)$ is the line are $y=2$ and $y=-2$ .

Therefore, the horizontal asymptote to the function $f\left(x\right)=\frac{2x-1}{\left|x\right|-3}$ are $y=2$ and $y=-2$ .