To determine the horizontal asymptote of the function
[tex]\[ y = \frac{6x + 18}{3x - 18}, \][/tex]
we need to consider the degrees of the polynomials in the numerator and the denominator.
1. Identify the degrees of the polynomials:
- The degree of the polynomial in the numerator [tex]\(6x + 18\)[/tex] is 1.
- The degree of the polynomial in the denominator [tex]\(3x - 18\)[/tex] is also 1.
2. Compare the degrees:
- Since the degrees of the numerator and the denominator are the same, the horizontal asymptote is determined by the ratio of the leading coefficients of these polynomials.
3. Find the leading coefficients:
- The leading coefficient of the numerator (the coefficient of the highest degree term [tex]\(x\)[/tex]) is 6.
- The leading coefficient of the denominator (the coefficient of the highest degree term [tex]\(x\)[/tex]) is 3.
4. Calculate the horizontal asymptote:
- The horizontal asymptote is obtained by dividing the leading coefficient of the numerator by the leading coefficient of the denominator:
[tex]\[
\frac{6}{3} = 2.
\][/tex]
Therefore, the horizontal asymptote of the function [tex]\[ y = \frac{6x + 18}{3x - 18} \][/tex] is
[tex]\[ y = 2. \][/tex]
