Chapter 2.2, Problem 19E

To determine

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

Answer to Problem 19E

The value of $\underset{x\to 0^{+}}{\lim }\left(\csc x\right)$ is $\infty$ .

Explanation of Solution

Given information:

The limit is $\underset{x\to 0^{+}}{\lim }\left(\csc x\right)$ .

Calculation:

Check the limit graphically.

To graph a function $y=\csc x$ , 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 $y=\csc x$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{1}\boxed{÷}\boxed{\text{SIN}}\boxed{\text{X}}\boxed{)}$ .

The display will show the equation,

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

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

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 Figure (1).

  

Figure (1)

As observed from graph, the function $f\left(x\right)=\csc x$ tends to $\infty$ as $x$ tends to $0$ from the right side.

Now check the limit by table.

To make the table of function $y=\csc x$ , 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 $y=\csc x$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{1}\boxed{÷}\boxed{\text{SIN}}\boxed{\text{X}}\boxed{)}$ .

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 $0.4$ , $0.3$ , $0.2$ , $0.1$ , $0.01$ , $0.001$ 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 right side.

Therefore, the value of $\underset{x\to 0^{+}}{\lim }\left(\csc x\right)$ is $\infty$ .