Chapter 1.2, Problem 34E

(a)

To determine

To find: The vertical asymptotes of the graph of. $f(x)$

Answer to Problem 34E

Vertical asymptotes at $x=n\pi$

Explanation of Solution

Given information: The fraction is $f(x)=\frac{\cot x}{\cos x}$

Calculation:

Rewrite $\cot x$ in terms of sine and cosine to simplify the rational statement. Any value that causes $\cos x=0$ will result in a hole rather than a vertical asymptote since $\cos x$ cancels. Since, $\sin x=0$ the vertical asymptotes will therefore appear at values of x. If you solve for, $x=0$ you get, $x=0,\pi ,2\pi$ etc. When $x=k\pi$ , and k is an integer, the vertical asymptotes are at multiples of $\pi$ .

  $\begin{array}{c}f(x)=\frac{\cos x/\sin x}{\cos x}\\ =\frac{\cos x}{\sin x\cos x}\\ =\frac{1}{\sin x}\end{array}$

Therefore the required vertical asymptotes at $x=k\pi$ ,where k is an integer.

(b)

To determine

To write: The behavior of $f(x)$ to the left and right of vertical asymptotes.

Answer to Problem 34E

  $\underset{x\to k{\pi }^{-}}{\lim }f(x)=\left\{\begin{array}{c}+\infty k\text{ is odd }\\ -\infty ,k\text{ is even }\end{array}\right.$

  $\underset{x\to k{\pi }^{+}}{\lim }f(x)=\left\{\begin{array}{c}-\infty ,k\text{ is odd }\\ +\infty ,k\text{ is even }\end{array}\right.$

Explanation of Solution

Given information: The fraction is $f(x)=\frac{\cot x}{\cos x}$

Graph:

The graph of the function is shown below:

  

The left- and right-limits at the vertical asymptote $x=k\pi$ were determined using a graphing tool and the knowledge that f is a periodic function with period. $2\pi$

  $\underset{x\to k{\pi }^{-}}{\lim }f(x)=\left\{\begin{array}{c}+\infty k\text{ is odd }\\ -\infty ,k\text{ is even }\end{array}\right.$

  $\underset{x\to k{\pi }^{+}}{\lim }f(x)=\left\{\begin{array}{c}-\infty ,k\text{ is odd }\\ +\infty ,k\text{ is even }\end{array}\right.$