Chapter 2.2, Problem 5E

(a)

To determine

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

Answer to Problem 5E

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

Explanation of Solution

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

Calculation:

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

The display will show the equation,

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

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 $f\left(x\right)=\frac{3x+1}{\left|x\right|+2}$ tends to $3$ as $x$ tends to $\infty$ .

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

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 $3$ as $x$ approaches to $\infty$ .

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

(b)

To determine

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

Answer to Problem 5E

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

Explanation of Solution

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

Calculation:

To find the limit of the function $f\left(x\right)=\frac{3x+1}{\left|x\right|+2}$ 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 $-3$ as $x$ tends to $-\infty$ .

To find the limit of the function $f\left(x\right)=\frac{3x+1}{\left|x\right|+2}$ 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 $-3$ 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 $f\left(x\right)=\frac{3x+1}{\left|x\right|+2}$ is $-3$ .

(c)

To determine

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

Answer to Problem 5E

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

Explanation of Solution

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

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 $f\left(x\right)=\frac{3x+1}{\left|x\right|+2}$ right hand limit is $\underset{x\to \infty }{\lim }f\left(x\right)=3$ and left hand limit is $\underset{x\to -\infty }{\lim }f\left(x\right)=-3$ . So, the horizontal asymptote to the curve $y=f\left(x\right)$ are $y=3$ and $y=-3$ .

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