To find the horizontal asymptote of the function

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

we need to analyze the degrees of the polynomial in the numerator and the denominator.

Step-by-Step Solution

1. Identify the Degrees:

   • The numerator $4x - x^3 - 2$ has a degree of 3 (the highest power of $x$ is $x^3$).
   • The denominator $2x^3 + x^2 - 3$ also has a degree of 3.

2. Determine the Horizontal Asymptote Based on Degree:

   • When the degrees of the numerator and the denominator are the same, the horizontal asymptote is determined by the ratio of the leading coefficients.

3. Extract the Leading Coefficients:

   • The leading coefficient of the numerator is $-1$ (from $-x^3$).
   • The leading coefficient of the denominator is $2$ (from $2x^3$).

4. Calculate the Horizontal Asymptote:

   • The horizontal asymptote is given by the ratio of the leading coefficients: $y = \frac{-1}{2} = -\frac{1}{2}.$

Final Answer

$y = -\frac{1}{2}.$

Would you like further details on this calculation or have any questions?

Here are five related questions for practice:

1. What is the horizontal asymptote if the degree of the numerator is higher than the degree of the denominator?
2. How would the horizontal asymptote change if the leading coefficients were positive?
3. What is the significance of horizontal asymptotes in real-world applications?
4. How do vertical asymptotes differ from horizontal asymptotes?
5. How would you find the oblique asymptote if the degrees were different?

Tip: Always start by comparing the degrees of the numerator and denominator to determine the behavior of rational functions at infinity.