Chapter 3.7, Problem 5CFU

To determine

To find: equations of vertical and horizontal asymptotes, if any, of the given function.

Answer to Problem 5CFU

  $\begin{array}{l}x=5\left[\text{Vertical asymptote}\right]\\ y=1\left[\text{Horizontal asymptote}\right]\end{array}$

Explanation of Solution

Given information:

  $f\left(x\right)=\frac{x}{x-5}$

Calculation:

The function $f\left(5\right)$ is undefined; there may be a vertical asymptote at $x=5$ . Lets make a table of values for the given function.

$x$	$f\left(x\right)$
4.9	$-$49
4.99	$-$499
4.999	$-$4999
4.9999	$-$49999

Since, the value in the table suggested that $f\left(x\right)\to -\infty$ as $x\to 5$ from left. So, the vertical asymptote is $x=5$ .

Solve $f\left(x\right)=y$ in terms of x as shown below to get the function where the function is undefined for value of y.

  $\begin{array}{c}y=\frac{x}{x-5}\\ y\left(x-5\right)=x\\ xy-5y=x\\ xy-x=5y\\ x\left(y-1\right)=5y\\ x=\frac{5y}{y-1}\end{array}$

The rational expression is undefined for $y=1$ . So, the horizontal asymptote is the line $y=1$ .