Find all vertical and horizontal asymptotes (if any). (If an answer does not exist, enter DNE. Enter your answers as a comma-separated list of equations.)
\[
r(x)=\frac{(x+1)(5 x-6)}{(x-5)(8 x+9)}
\]
vertical asymptote(s)
horizontal asymptote
Answer
\(\boxed{\text{The vertical asymptotes of the function are } x = 5 \text{ and } x = -\frac{9}{8} \text{ and the horizontal asymptote is } y = \frac{5}{8}. \text{ There are no oblique asymptotes for this function.}}\)
Step 1 ：Set the denominator of the function equal to zero and solve for x to find the vertical asymptote: \((x-5)(8x + 9) = 0\) which gives \(x = 5\) and \(x = -\frac{9}{8}\).
Step 2 ：Compare the degrees of the numerator and the denominator to find the horizontal asymptote. Since the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients, which is \(\frac{5}{8}\).
Step 3 ：Check if there is an oblique asymptote. Since the degrees of the numerator and the denominator are the same, there is no oblique asymptote.
Step 4 ：\(\boxed{\text{The vertical asymptotes of the function are } x = 5 \text{ and } x = -\frac{9}{8} \text{ and the horizontal asymptote is } y = \frac{5}{8}. \text{ There are no oblique asymptotes for this function.}}\)
