Chapter 2.2, Problem 18E

To determine

To find: The value of $\underset{x\to 0^{-}}{\lim }\left(\frac{\mathrm{int}x}{x}\right)$ with the help of graph and table.

Answer to Problem 18E

The value of $\underset{x\to 0^{-}}{\lim }\left(\frac{\mathrm{int}x}{x}\right)$ is $\infty$ .

Explanation of Solution

Given information: The limit is $\underset{x\to 0^{-}}{\lim }\left(\frac{\mathrm{int}x}{x}\right)$ .

Calculation:

Find the limit graphically.

To graph a function $y=\frac{\mathrm{int}x}{x}$ , 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 $y=\frac{\mathrm{int}x}{x}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{MATH}}\boxed{\to }\boxed{5}\boxed{\text{X}}\boxed{)}\boxed{÷}\boxed{\text{X}}$ .

The display will show the equation,

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

Press the window key and adjust the window to $\left[-4,4\right]$ by $\left[-3,3\right]$ .

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 $f\left(x\right)=\frac{\mathrm{int}x}{x}$ tends to $\infty$ as $x$ tends to $0$ from the left side.

Now, find the limit by table.

To make the table of function $y=\frac{\mathrm{int}x}{x}$ , 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 $y=\frac{\mathrm{int}x}{x}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{MATH}}\boxed{\to }\boxed{5}\boxed{\text{X}}\boxed{)}\boxed{÷}\boxed{\text{X}}$ .

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

To draw the table, enter the keystrokes $\boxed{2\text{nd}}\boxed{\text{GRAPH}}$ , and then enter the values of $\text{X}$ as $-0.4$ , $-0.3$ , $-0.2$ , $-0.1$ , $-0.01$ , $-0.001$ and $-0.0001$ . The table is shown in window.

  

Figure (2)

As observed from table the value of $\text{Y}$ approaches $\infty$ as $x$ approaches to $0$ from the left side.

Therefore, the value of $\underset{x\to 0^{-}}{\lim }\left(\frac{\mathrm{int}x}{x}\right)$ is $\infty$ .