To find the horizontal asymptote of the function

$f(x) = \frac{2x + x^3 + 4}{-5x^3 - 4x^2 + 3},$

we need to examine the degrees of the polynomials in the numerator and denominator.

1. Degree of the numerator: The highest power of $x$ in the numerator $2x + x^3 + 4$ is $x^3$, so the degree of the numerator is 3.
2. Degree of the denominator: The highest power of $x$ in the denominator $-5x^3 - 4x^2 + 3$ is also $x^3$, so the degree of the denominator is also 3.

When the degrees of the numerator and denominator are the same, the horizontal asymptote is found by dividing the leading coefficients of the highest degree terms from both the numerator and the denominator.

• The leading coefficient in the numerator is $1$ (from $x^3$).
• The leading coefficient in the denominator is $-5$ (from $-5x^3$).

Thus, the horizontal asymptote is:

$y = \frac{1}{-5} = -\frac{1}{5}.$

So, the horizontal asymptote is:

$y = -\frac{1}{5}.$

Would you like more details or have any questions?

Here are five related questions:

1. How do you find the horizontal asymptote when the degree of the numerator is less than that of the denominator?
2. What happens if the degree of the numerator is greater than the degree of the denominator?
3. Can rational functions have oblique (slant) asymptotes?
4. How do vertical asymptotes differ from horizontal asymptotes?
5. What are some real-world applications of asymptotes in mathematical modeling?

Tip: Always check the degrees of the numerator and denominator when finding horizontal asymptotes, as this relationship determines whether one exists and where it is located.