\( \mid< \) Find the horizontal asymptote, if any, of the graph of the rational function. - \( g(x)=\frac{8 x^{2}}{2 x^{2}+1} \)

To find the horizontal asymptote of the function \( g(x) = \frac{8x^{2}}{2x^{2}+1} \), we look at the degrees of the polynomial in the numerator and the denominator. Both the numerator and the denominator are polynomials of degree 2. In such cases, the horizontal asymptote is found by taking the ratio of the leading coefficients. Here, the leading coefficient of the numerator (8) over the leading coefficient of the denominator (2) gives us \( \frac{8}{2} = 4 \). Thus, the horizontal asymptote is \( y = 4 \). Additionally, as \( x \) approaches infinity or negative infinity, the values of \( g(x) \) will get closer and closer to 4, confirming that the function does indeed approach this line as it extends horizontally.