Solution:

The function is:

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

To determine the horizontal asymptote of a rational function, we analyze the degrees of the numerator and denominator:

1. Degree of the numerator: $\deg(5x^2 + 3) = 2$,
2. Degree of the denominator: $\deg(5x^2 - 3) = 2$.

Since the degrees of the numerator and denominator are equal, the horizontal asymptote is given by:

$y = \frac{\text{leading coefficient of the numerator}}{\text{leading coefficient of the denominator}}.$

In this case, the leading coefficient of both the numerator and the denominator is $5$. Therefore:

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

Final Answer:

The horizontal asymptote is:

$y = 1$ (Option A).

Would you like further explanation? Here are some related questions to consider:

1. What happens to the function as $x \to \infty$ or $x \to -\infty$?
2. How do horizontal asymptotes differ from vertical asymptotes?
3. What if the numerator had a higher degree than the denominator?
4. How would the graph of this function behave near the asymptotes?
5. Could there be oblique (slant) asymptotes for this function?

Tip:

Horizontal asymptotes describe the behavior of a function as $x \to \pm\infty$. They are determined solely by the highest degree terms in the numerator and denominator!