To solve the problem and determine the horizontal asymptote of the rational function:

$g(x) = \frac{10x^2}{5x^2 + 1}$

Step-by-Step Approach:

1. Degrees of the Polynomials: The horizontal asymptote of a rational function depends on the degrees of the numerator and the denominator.

   • The degree of the numerator $10x^2$ is 2.
   • The degree of the denominator $5x^2 + 1$ is also 2.

2. Comparison of Degrees:

   • If the degree of the numerator equals the degree of the denominator, the horizontal asymptote is given by the ratio of the leading coefficients.

3. Leading Coefficients:

   • The leading coefficient of the numerator is 10.
   • The leading coefficient of the denominator is 5.

4. Horizontal Asymptote:

   • Since the degrees are the same, the horizontal asymptote is: $y = \frac{10}{5} = 2$

Thus, the horizontal asymptote is $y = 2$.

The correct choice would be:

• A. The horizontal asymptote is $y = 2$.

Do you want further details or have any questions?
Here are 5 related questions to expand on this topic:

1. What happens to the horizontal asymptote if the degree of the numerator is less than the denominator?
2. How do vertical asymptotes differ from horizontal asymptotes?
3. Can rational functions have both vertical and horizontal asymptotes?
4. What is the general rule for determining slant (oblique) asymptotes?
5. How would you find the intercepts of a rational function?

Tip: Always compare the degrees of the numerator and denominator when determining horizontal asymptotes.