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Find the horizontal asymptote.

[tex]\[ y = \frac{6x - 18}{x + 9} \][/tex]

[tex]\[ y = [?] \][/tex]

Sagot :

GinnyAnswer
To find the horizontal asymptote of the rational function [tex]\(y = \frac{6x - 18}{x + 9}\)[/tex], follow these steps:

1. Identify the degree of the numerator and the denominator:
- The numerator is [tex]\(6x - 18\)[/tex], which is a polynomial of degree 1.
- The denominator is [tex]\(x + 9\)[/tex], which is also a polynomial of degree 1.

2. Determine the leading terms of the numerator and the denominator:
- The leading term in the numerator is [tex]\(6x\)[/tex].
- The leading term in the denominator is [tex]\(x\)[/tex].

3. Divide the leading coefficients of the highest degree terms:
- The coefficient of [tex]\(x\)[/tex] in the numerator is [tex]\(6\)[/tex].
- The coefficient of [tex]\(x\)[/tex] in the denominator is [tex]\(1\)[/tex].

4. Find the horizontal asymptote by dividing these coefficients:
[tex]\[ \text{Horizontal Asymptote} = \frac{\text{Numerator's leading coefficient}}{\text{Denominator's leading coefficient}} = \frac{6}{1} = 6 \][/tex]

So, the horizontal asymptote of the function [tex]\(y = \frac{6x - 18}{x + 9}\)[/tex] is [tex]\(y = 6\)[/tex].
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