Chapter 1.2, Problem 32E

(a)

To determine

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

Answer to Problem 32E

Vertical asymptotes at $x=\frac{\pi }{2}+\pi k$

Explanation of Solution

Given information: The fraction is $f(x)=\sec x$

Calculation:

The definition of the function $\sec x$ is:

  $\sec (x)=\frac{1}{\cos (x)}$

In a section where the function is unknowable, vertical asymptotes will exist. Since $\cos (x)=0$ in this situation, the function is undefined. only when x is represented by

  $x=\frac{\pi }{2}+\pi k$

where k is an integer. The function $\sec x$ will have infinitely many asymptotes since k can be any whole number.

(b)

To determine

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

Answer to Problem 32E

When k is even:

The left hand limit is $\infty$

The right hand limit is $-\infty$

When k is odd:

The left hand limit is $-\infty$

The right hand limit is $\infty$

Explanation of Solution

Given information: The fraction is $f(x)=\sec x$

Graph:

The graph of the function is shown below:

  

From the graph:

Every vertical asymptote is located at $x=\pi /2+k\pi$ where k is an integer.

When k is even (Example: $k=2,x=\pi /2+2\pi =5\pi /2$ )

The left hand limit is $\infty$

The right hand limit is $-\infty$

When k is odd (Example: $k=1,x=\pi /2+\pi =3\pi /2$ )

The left hand limit is $-\infty$

The right hand limit is $\infty$