Chapter 3.7, Problem 22E

To determine

To find the vertical and horizontal asymptote of the rational function.

Answer to Problem 22E

The vertical asymptote are the lines $x=1$ and $x=-1$ . and the horizontal asymptote is the line $y=0$ , of the given rational function.

Explanation of Solution

Given information:

The rational function is $r\left(x\right)=\frac{2x-3}{x^2-1}$

Calculation:

Vertical asymptote of $r\left(x\right)$ are the lines $x=a$ , where $a$ is the zero of the denominator.

Therefore,

The zero of the denominators is obtained as:

  $\begin{array}{l}{x}^2-1=0\\ \left(x+1\right)\left(x-1\right)=0\end{array}$

Either,

  $x=1$

Or,

  $x=-1$

Hence,

The vertical asymptote are the lines $x=1$ and $x=-1$ .

Horizontal asymptote:

The degree of the numerator is 1 and denominator is 2.

Since, $n=1<m=2$ , so, horizontal asymptote of the rational function $r\left(x\right)$ is the line $y=0$ .

Hence,

The vertical asymptote are the lines $x=1$ and $x=-1$ and the horizontal asymptote is the line $y=0$ , of the given rational function.