To determine the horizontal asymptote of the function [tex]\( y = \frac{3x + 12}{x - 6} \)[/tex], we need to find the behavior of the function as [tex]\( x \)[/tex] approaches infinity. Essentially, we're interested in the limit of the function [tex]\( y \)[/tex] as [tex]\( x \)[/tex] goes to infinity.
Here's a step-by-step approach:
1. Rewrite the function in a form that highlights the behavior as [tex]\( x \)[/tex] becomes very large:
[tex]\[
y = \frac{3x + 12}{x - 6}
\][/tex]
2. Divide the numerator and the denominator by [tex]\( x \)[/tex], the highest power of [tex]\( x \)[/tex] present in the function:
[tex]\[
y = \frac{3x + 12}{x - 6} = \frac{3 + \frac{12}{x}}{1 - \frac{6}{x}}
\][/tex]
3. Analyze the terms involving [tex]\( \frac{12}{x} \)[/tex] and [tex]\( \frac{6}{x} \)[/tex]:
- As [tex]\( x \)[/tex] approaches infinity, [tex]\( \frac{12}{x} \)[/tex] approaches 0.
- Similarly, [tex]\( \frac{6}{x} \)[/tex] also approaches 0.
4. Simplify the function using these limits:
[tex]\[
y = \frac{3 + \frac{12}{x}}{1 - \frac{6}{x}} \approx \frac{3 + 0}{1 - 0} = \frac{3}{1} = 3
\][/tex]
Thus, as [tex]\( x \)[/tex] approaches infinity or negative infinity, the function [tex]\( y = \frac{3x + 12}{x - 6} \)[/tex] approaches the value 3. This value is the horizontal asymptote of the function.
Therefore, the horizontal asymptote of the function [tex]\( y = \frac{3x + 12}{x - 6} \)[/tex] is:
[tex]\[
y = 3
\][/tex]
