Chapter 1.2, Problem 31E

(a)

To determine

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

Answer to Problem 31E

Vertical asymptotes at $x=n\pi$

Explanation of Solution

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

Calculation:

The line $x=a$ is a vertical asymptote for $y=f(x)$ if at least one of the following is true:

  $\begin{array}{c}\underset{x\to a^{-}}{\lim }f(x)=\infty \\ \underset{x\to a^{+}}{\lim }f(x)=\infty \\ \underset{x\to a^{-}}{\lim }f(x)=-\infty \\ \underset{x\to a^{+}}{\lim }f(x)=-\infty \end{array}$

The given function is:

  $y=\cot x$

This can be rewritten as

  $y=\frac{\cos x}{\sin x}$

(1) For all real numbers, the numerator of $\cos x$ is finite.

(2) When x is an integer multiple of $\pi$ , the denominator is 0.

(3) When $\cos x$ is 0, the numerator is non-zero.

As a result $x=n\pi$ , is the vertical asymptotes.

(b)

To determine

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

Answer to Problem 31E

The left hand limit is $-\infty$ and the right hand limit is $+\infty$

Explanation of Solution

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

Graph:

The graph of the function is shown below:

  

The function $f(x)=\cot x$ has infinitely many asymptotes and each asymptote is of form $x=k\pi$ where k is a whole number.(On picture, the asymptotes are red vertical lines)

Now we can see that if get closer to any of the asymptotes from the left side then the function tends to go $-\infty$ and if get closer from the right side of any asymptote then the function tends to go to $+\infty$ .

The left hand limit is $-\infty$ and the right hand limit is $+\infty$