Chapter 2.2, Problem 8E

(a)

To determine

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

Answer to Problem 8E

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

Explanation of Solution

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

Calculation:

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

The display will show the equation,

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

Press the window key and adjust the window to $\left[-5,5\right]$ by $\left[-2,2\right]$ .Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown below.

  

Figure (1)

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

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

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 $100$ , $200$ , $500$ , $1000$ , $-100$ , $-200$ and $-500$ . The table is shown in window.

  

Figure (2)

As observed from table the value of $\text{Y}$ approaches to $1$ as $x$ approaches to $\infty$ .

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

(b)

To determine

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

Answer to Problem 8E

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

Explanation of Solution

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

Calculation:

To find the limit of the function $f\left(x\right)=\frac{\left|x\right|}{\left|x\right|+1}$ 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 $1$ as $x$ tends to $-\infty$ .

To find the limit of the function $f\left(x\right)=\frac{\left|x\right|}{\left|x\right|+1}$ 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 $1$ 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 $f\left(x\right)=\frac{\left|x\right|}{\left|x\right|+1}$ is $1$ .

(c)

To determine

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

Answer to Problem 8E

The horizontal asymptote to the function $f\left(x\right)=\frac{\left|x\right|}{\left|x\right|+1}$ is $y=1$ .

Explanation of Solution

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

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 $f\left(x\right)=\frac{\left|x\right|}{\left|x\right|+1}$ right hand limit is $\underset{x\to \infty }{\lim }f\left(x\right)=1$ and left hand limit is $\underset{x\to -\infty }{\lim }f\left(x\right)=1$ . So, the horizontal asymptote to the curve $y=f\left(x\right)$ is the line $y=1$ .

Therefore, the horizontal asymptote to the function $f\left(x\right)=\frac{\left|x\right|}{\left|x\right|+1}$ is $y=1$ .