Find the horizontal and vertical asymptote(s) of $LaTeX: \displaystyle f(x) = \frac{x^{2} - 19 x + 88}{9 x^{2} - 18 x - 135}$ , if they exist.

The degree of the numerator is the same as the degree of the denominator, so the horizontal asymptote is the ratio of the leading coefficients $LaTeX: \displaystyle y=\frac{1}{9}$ . The vertical asymptotes are the zeros of the denominator. $LaTeX: \displaystyle 9 x^{2} - 18 x - 135=0 \iff 9 \left(x - 5\right) \left(x + 3\right)=0$ . The vertical asymptotes are $LaTeX: \displaystyle x=-3$ and $LaTeX: \displaystyle x=5$ .