Chapter 2.2, Problem 31E

(a)

To determine

To find: The vertical asymptotes of the graph of $f\left(x\right)=\cot x$ .

Answer to Problem 31E

The vertical asymptotes of the graph of $f\left(x\right)=\cot x$ are $x=k\pi$ .

Explanation of Solution

Given information:The function is $f\left(x\right)=\cot x$ .

Calculation:

The function can be rewritten as:

  $\begin{array}{c}y=\cot x\\ =\frac{1}{\tan x}\end{array}$

To graph a function $y=\frac{1}{\tan x}$ , 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 $y=\cot x$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{1}\boxed{÷}\boxed{(}\boxed{\text{TAN}}\boxed{\text{X}}\boxed{)}\boxed{)}$ .

The display will show the equation,

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

Press the window key and adjust the window to $\left[-2\pi ,2\pi \right]$ 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 below.

  

Figure (1)

As observed from graph of function $f\left(x\right)=\cot x$ , the vertical asymptotes to the graph are of type $x=k\pi$ for any integer $k$ .

Therefore, the vertical asymptotes of the graph of $f\left(x\right)=\cot x$ are $x=k\pi$ .

(b)

To determine

To find:The behavior of the function $f\left(x\right)=\cot x$ to the right and left of the vertical asymptotes.

Answer to Problem 31E

The value of $\underset{x\to k{\pi }^{-}}{\lim }f\left(x\right)$ , $\underset{x\to k{\pi }^{+}}{\lim }f\left(x\right)$ are $-\infty$ and $\infty$ respectively.

Explanation of Solution

Given information:The function is $f\left(x\right)=\cot x$ .

Calculation:

As observed from the graph of the given function drawn in part (a), the left hand limit for function at each point $x=k\pi$ is $-\infty$ . So, the value of $\underset{x\to k{\pi }^{-}}{\lim }f\left(x\right)$ is $-\infty$ .

As observed from the graph of the given function drawn in part (a), the right hand limit for function at each point $x=k\pi$ is $\infty$ .So, the value of $\underset{x\to k{\pi }^{+}}{\lim }f\left(x\right)$ is $\infty$ .

Therefore, the value of $\underset{x\to k{\pi }^{-}}{\lim }f\left(x\right)$ , $\underset{x\to k{\pi }^{+}}{\lim }f\left(x\right)$ are $-\infty$ and $\infty$ respectively.