To find the horizontal asymptotes of the function \( f(x) = \frac{x^2 - 5x + 6}{x^3 - 8} \), we analyze the degrees of the polynomial in the numerator and the polynomial in the denominator.
1. Determine the degrees of the numerator and denominator:
- The degree of the numerator \( x^2 - 5x + 6 \) is 2 (since the highest power of \( x \) is \( x^2 \)).
- The degree of the denominator \( x^3 - 8 \) is 3 (since the highest power of \( x \) is \( x^3 \)).
2. Compare the degrees to determine the horizontal asymptote:
- If the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote at \( y = 0 \).
- If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is \( y = \frac{\text{leading coefficient of the numerator}}{\text{leading coefficient of the denominator}} \).
- If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (though there may be an oblique asymptote).
In this case:
- The degree of the numerator (2) is less than the degree of the denominator (3).
Therefore, the horizontal asymptote of the function \( f(x) \) is:
[tex]\[ y = 0 \][/tex]
This completes the analysis and identifies the horizontal asymptote for the given function [tex]\( f(x)=\frac{x^2-5 x+6}{x^3-8} \)[/tex]. The horizontal asymptote is [tex]\( y = 0 \)[/tex].
