Chapter 1.2, Problem 27E

(a)

To determine

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

Answer to Problem 27E

Vertical asymptotes at

  $x=2$ and

  $x=-2$

Explanation of Solution

Given information: The fraction is $f(x)=\frac{1}{x^2-4}$

Calculation:

Any value of $x$ that causes the denominator to equal zero will cause the function to have vertical asymptotes. By solving for $x$ while holding the denominator of the expression $x^2-4$ equal to 0, the following result is produced:

  $\begin{array}{c}{x}^2-4=0\\ \Rightarrow x^2=4\\ \Rightarrow x=\pm \sqrt{4}\\ =\pm 2\end{array}$

The graph will then have vertical asymptotes at $x=2$

And $x=-2$ .

(b)

To determine

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

Answer to Problem 27E

The left-hand limit of the function at -2 is $-\infty$

The right-hand limit of the function a 2 is $\infty$

The left-hand limit of the function at 2 is $-\infty$

The right-hand limit of the function at 2 is $\infty$

Explanation of Solution

Given information: The fraction is $f(x)=\frac{1}{x^2-4}$

Calculation:

The graph of the function is shown below:

  

The left-hand limit of the function at -2 is $-\infty$

The right-hand limit of the function a 2 is $\infty$

The left-hand limit of the function at 2 is $-\infty$

The right-hand limit of the function at 2 is $\infty$