Chapter 8.3, Problem 5LC

Solution

To find: The vertical asymptotes of the rational function $y=\frac{x-3}{x+5}$ .

The vertical asymptote is at $x=-5$ .

Given:

The rational function is $y=\frac{x-3}{x+5}$ .

Concept:

The rational function $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$ has vertical asymptote at each real zero of $q\left(x\right)$ , if $p\left(x\right)$ and $q\left(x\right)$ have no common zeros.

Calculation:

Since the rational function $y=\frac{x-3}{x+5}$ is already in its reduced form, because the numerator and denominator of the rational function has no common factors or zeros in it.

Therefore, with the help of concept stated above, the vertical asymptote of the function from the reduced form $\left(1\right)$ is evaluated as:

  $\begin{array}{c}y=\frac{x-3}{x+5}\mathrm{....}\left(\text{Reduced form}\right)\\ x+5=0\mathrm{....}\left(\text{Set denominator equal to zero}\right)\\ x=-5\end{array}$

Hence, the vertical asymptote of the rational function is at $x=-5$ .

Conclusion:

The vertical asymptote is at $x=-5$ .