Chapter 13.4, Problem 19E

To determine

To find: The value of $\underset{x\to -\infty }{\lim }\left(\frac{\sqrt{x^2+4x}}{4x+1}\right)$ with the help of table of values and then confirm the answer graphically.

Answer to Problem 19E

The value of $\underset{x\to -\infty }{\lim }\left(\frac{\sqrt{x^2+4x}}{4x+1}\right)$ is equal to $-0.25$ .

Explanation of Solution

Given information:

The expression is $\underset{x\to -\infty }{\lim }\left(\frac{\sqrt{x^2+4x}}{4x+1}\right)$ .

Calculation:

Now check the limit by table.

To make the table of function $y=\frac{\sqrt{x^2+4x}}{4x+1}$ , 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 $y=\frac{\sqrt{x^2+4x}}{4x+1}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{2nd}}\boxed{x^2}\boxed{\text{X}}\boxed{^}\boxed{2}\boxed{+}\boxed{4}\boxed{\text{X}}\boxed{)}\boxed{÷}\boxed{(}\boxed{4}\boxed{\text{X}}\boxed{+}\boxed{1}\boxed{)}$ .

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 $\boxed{2\text{nd}}\boxed{\text{GRAPH}}$ , Enter the values of $\text{X}$ as $-1$ , $-10$ , $-100$ , $-1000$ , $-10000$ and $-100000$ . The table is shown in window.

  

Figure(1)

As observed from table the value of $\text{Y}$ approaches $-0.25$ as $x$ approaches to $-\infty$ .

Now check the limit graphically.

To graph a function $y=\frac{\sqrt{x^2+4x}}{4x+1}$ , 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 $y=\frac{\sqrt{x^2+4x}}{4x+1}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{2nd}}\boxed{x^2}\boxed{\text{X}}\boxed{^}\boxed{2}\boxed{+}\boxed{4}\boxed{\text{X}}\boxed{)}\boxed{÷}\boxed{(}\boxed{4}\boxed{\text{X}}\boxed{+}\boxed{1}\boxed{)}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=\sqrt{\left(\text{X}^2+4\text{X}\right)}/\left(4\text{X}+1\right)$

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 (2).

  

Figure (2)

As observed from graph, the function $y=\frac{\sqrt{x^2+4x}}{4x+1}$ tends to $-0.25$ as $x$ tends to $-\infty$ .

Therefore, the value of $\underset{x\to -\infty }{\lim }\left(\frac{\sqrt{x^2+4x}}{4x+1}\right)$ is equal to $-0.25$ .