Chapter 1.2, Problem 28E

(a)

To determine

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

Answer to Problem 28E

Vertical asymptotes at

  $x=-2$

Explanation of Solution

Given information: The fraction is $y=\frac{x^2-1}{2x+4}$

Calculation:

The numerator and denominator have no factor in common and there are no removal discontinuities.

The graph of $f(x)$ will have vertical asymptotes where the denominators equals zero.

  $\begin{array}{c}2x+4=0\\ \to x=-2\end{array}$

Therefore the required vertical asymptotes at $x=-2$ .

(b)

To determine

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

Answer to Problem 28E

There is a vertical asymptote at $x=-2$ due to the infinite nature of the left-hand limit and right-hand limits.

Explanation of Solution

Given information: The fraction is $y=\frac{x^2-1}{2x+4}$

Graph:

The graph of the function is shown below:

  

From the graph:

  $\underset{x\to -2^{-}}{\lim }f(x)=-\infty$

  $\underset{x\to -2^{+}}{\lim }f(x)=\infty$

There is a vertical asymptote at $x=-2$ due to the infinite nature of the left-hand limit and right-hand limits.