To find the horizontal asymptote of the function [tex]\( y = \frac{3x + 30}{x - 10} \)[/tex], we need to analyze the behavior of the function as [tex]\( x \)[/tex] approaches positive or negative infinity.
1. Identify the degree of the polynomials in the numerator and the denominator:
- The numerator, [tex]\( 3x + 30 \)[/tex], is a linear polynomial of degree 1.
- The denominator, [tex]\( x - 10 \)[/tex], is also a linear polynomial of degree 1.
2. When the degrees of the numerator and the denominator are the same, the horizontal asymptote can be determined by the ratio of the leading coefficients.
- The leading coefficient of the numerator is 3.
- The leading coefficient of the denominator is 1.
3. The horizontal asymptote [tex]\( y \)[/tex] is the ratio of these leading coefficients:
[tex]\[
y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}} = \frac{3}{1} = 3
\][/tex]
Thus, the horizontal asymptote of the function [tex]\( y = \frac{3x + 30}{x - 10} \)[/tex] is [tex]\( y = 3 \)[/tex].
