The function provided is:

$f(x) = \frac{1 - 5x - 2x^2}{3x^2 + 7}$

The goal is to find the horizontal asymptote of this function as $x \to \infty$.

Step-by-step analysis:

1. Degrees of numerator and denominator:

   • The degree of the numerator is 2 (since the highest power of $x$ is $x^2$).
   • The degree of the denominator is also 2 (since the highest power of $x$ is also $x^2$).

2. Horizontal Asymptote Rule:

   • If the degrees of the numerator and the denominator are the same, the horizontal asymptote is given by the ratio of the leading coefficients of the highest-degree terms.
   • The leading term of the numerator is $-2x^2$, and the leading term of the denominator is $3x^2$.

   Therefore, the horizontal asymptote is:

   $y = \frac{-2}{3}$

Thus, the horizontal asymptote of the function is $y = \frac{-2}{3}$.

Let me know if you'd like further details or have any questions!

Here are 5 related questions for further practice:

1. How do you find vertical asymptotes of a rational function?
2. What happens if the degree of the numerator is greater than the degree of the denominator?
3. How would you determine slant asymptotes in rational functions?
4. Can a function have more than one horizontal asymptote?
5. How does the behavior of the function near an asymptote affect the graph?

Tip: For rational functions, always compare the degrees of the numerator and denominator to quickly determine horizontal asymptotes.