To find the equation of the horizontal asymptote of the function [tex]\( g(x) = \frac{7x^2}{x^2 + 5} \)[/tex], we need to analyze the degrees of the polynomial in the numerator and the polynomial in the denominator.
1. Identify the degrees of the polynomials:
- The degree of the numerator, [tex]\( 7x^2 \)[/tex], is 2.
- The degree of the denominator, [tex]\( x^2 + 5 \)[/tex], is also 2.
2. Compare the degrees:
- Since the degrees of both the numerator and the denominator are the same, the horizontal asymptote can be found by dividing the leading coefficients of the numerator and the denominator.
3. Determine the leading coefficients:
- The leading coefficient of the numerator [tex]\( 7x^2 \)[/tex] is 7.
- The leading coefficient of the denominator [tex]\( x^2 + 5 \)[/tex] is 1.
4. Calculate the horizontal asymptote:
- Divide the leading coefficient of the numerator by the leading coefficient of the denominator:
[tex]\[
\frac{7}{1} = 7
\][/tex]
Therefore, the horizontal asymptote of the function [tex]\( g(x) = \frac{7x^2}{x^2 + 5} \)[/tex] is [tex]\( y = 7 \)[/tex].
Thus, the correct answer is:
C. [tex]\( y = 7 \)[/tex]
