To determine the horizontal asymptote of the given function

$f(x) = \frac{4x^2 - 1}{5x + 3},$

we need to examine the behavior of $f(x)$ as $x$ approaches infinity or negative infinity.

Step 1: Determine the degrees of the polynomial.

• The degree of the numerator $4x^2 - 1$ is 2, since $4x^2$ is the highest power term.
• The degree of the denominator $5x + 3$ is 1, as $5x$ is the highest power term.

Step 2: Apply the horizontal asymptote rule.

For rational functions, the horizontal asymptote depends on the degrees of the polynomials in the numerator and denominator:

1. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$.
2. If the degree of the numerator equals the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients.
3. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (the function may have an oblique asymptote).

In this case, the degree of the numerator (2) is greater than the degree of the denominator (1). Therefore, there is no horizontal asymptote for this function. Instead, this function will have an oblique (slant) asymptote.

Let me know if you want more details about finding the oblique asymptote.

Do you want more details or have any questions?

Here are 5 related questions for further exploration:

1. How do you find the oblique asymptote for this function?
2. What is the significance of an oblique asymptote in graphing a rational function?
3. How do you determine vertical asymptotes for rational functions like this?
4. How can you find the behavior of the function near its asymptotes?
5. What are the conditions under which a rational function has no asymptote at all?

Tip: In rational functions, always compare the degrees of the numerator and denominator to decide which type of asymptote exists.