Chapter 2.2, Problem 20E

To determine

To find: The value of $\underset{x\to {\frac{\pi }{2}}^{+}}{\lim }\left(\sec x\right)$ with the help of graph and table.

Answer to Problem 20E

The value of $\underset{x\to {\frac{\pi }{2}}^{+}}{\lim }\left(\sec x\right)$ is $-\infty$ .

Explanation of Solution

Given information:

The limit is $\underset{x\to {\frac{\pi }{2}}^{+}}{\lim }\left(\sec x\right)$ .

Calculation:

Check the limit graphically.

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

The display will show the equation,

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

Press the window key and adjust the window to $\left[-\pi ,\pi \right]$ 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 $f\left(x\right)=\sec x$ tends to $-\infty$ as $x$ tends to $\frac{\pi }{2}$ from the right side.

Now check the limit by table.

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

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 $1.6$ , $1.59$ , $1.58$ , $1.579$ , $1.576$ , $1.574$ and $1.572$ . The table is shown in window.

  

Figure(2)

As observed from table the value of $\text{Y}$ approaches $-\infty$ as $x$ approaches to $\frac{\pi }{2}$ from the right side.

Therefore, the value of $\underset{x\to {\frac{\pi }{2}}^{+}}{\lim }\left(\sec x\right)$ is $-\infty$ .