College Algebra, Chapter 4, 4.6, Section 4.6, Problem 28

Find all horizontal and vertical asymptotes of the rational function $\displaystyle r(x) = \frac{(2x - 1)(x + 3)}{(3x - 1)(x - 4)}$.

Since the degree of the numerator is equal to the degree of the denominator, then the horizontal asymptote = $\displaystyle \frac{\text{leading coefficient of the numerator}}{\text{leading coefficient of the denominator}} = \frac{2}{3}$. Thus, the horizontal asymptote is $\displaystyle y = \frac{2}{3}$.

To determine the vertical asymptotes, we set the denominator equal to .


$
\begin{equation}
\begin{aligned}

(3x - 1)(x - 4) =& 0
&&
\\
\\
3x - 1 =& 0 \text{ and } x - 4 = 0
&& \text{Zero Product Property}

\end{aligned}
\end{equation}
$


Thus, the vertical asymptotes are $\displaystyle x = \frac{1}{3}$ and $\displaystyle x = 4$.