Find the horizontal and vertical asymptote(s) of $LaTeX: \displaystyle f(x) = \frac{x^{2} + 3 x - 10}{7 x^{2} + 63 x + 126}$ , 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 $LaTeX: \displaystyle y=\frac{1}{7}$ . The vertical asymptotes are the zeros of the denominator. $LaTeX: \displaystyle 7 x^{2} + 63 x + 126=0 \iff 7 \left(x + 3\right) \left(x + 6\right)=0$ . The vertical asymptotes are $LaTeX: \displaystyle x=-6$ and $LaTeX: \displaystyle x=-3$ .