To determine the horizontal asymptote of the function
[tex]\[ g(x) = \frac{x^3 - 5x^2 + x - 3}{x^2 - 13}, \][/tex]
we need to analyze the degrees of the polynomial in the numerator and the denominator.
1. Degree of the Numerator:
- The numerator is [tex]\( x^3 - 5x^2 + x - 3 \)[/tex].
- The highest power of [tex]\( x \)[/tex] in the numerator is [tex]\( x^3 \)[/tex], hence the degree of the numerator is 3.
2. Degree of the Denominator:
- The denominator is [tex]\( x^2 - 13 \)[/tex].
- The highest power of [tex]\( x \)[/tex] in the denominator is [tex]\( x^2 \)[/tex], hence the degree of the denominator is 2.
3. Comparison of Degrees:
- The degree of the numerator (3) is greater than the degree of the denominator (2).
4. Horizontal Asymptote Rules:
- When the degree of the numerator is greater than the degree of the denominator, the function does not have a horizontal asymptote. Instead, the function grows without bound as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] or [tex]\( -\infty \)[/tex].
Based on this analysis, it is concluded that:
There is no horizontal asymptote for the function [tex]\( g(x) \)[/tex].
