To find the horizontal asymptote of the function
[tex]\[
y = \frac{4x + 12}{4x - 12}
\][/tex]
we follow a systematic approach:
1. Identify the degrees of the polynomials:
- The degree of the numerator [tex]\(4x + 12\)[/tex] is 1 (since [tex]\(4x\)[/tex] is a linear term).
- The degree of the denominator [tex]\(4x - 12\)[/tex] is also 1 (since [tex]\(4x\)[/tex] is a linear term).
2. Compare the degrees:
- When the degree of the numerator and the degree of the denominator are the same, the horizontal asymptote is found by taking the ratio of the leading coefficients.
3. Extract the leading coefficients:
- For the numerator [tex]\(4x + 12\)[/tex], the leading coefficient is 4.
- For the denominator [tex]\(4x - 12\)[/tex], the leading coefficient is also 4.
4. Form the ratio:
- The horizontal asymptote is given by the ratio of the leading coefficients of the numerator and the denominator.
[tex]\[
\frac{\text{Leading coefficient of the numerator}}{\text{Leading coefficient of the denominator}} = \frac{4}{4} = 1
\][/tex]
Therefore, the horizontal asymptote of the function [tex]\( y = \frac{4x + 12}{4x - 12} \)[/tex] is:
[tex]\[
y = 1
\][/tex]
