Questions: Determine the horizontal asymptote of the graph of the function. g(x)=(x^3-5x^2+x-1)/(x^2-15)

Transcript text: Determine the horizontal asymptote of the graph of the function. \[ g(x)=\frac{x^{3}-5 x^{2}+x-1}{x^{2}-15} \]

Solution

Step 1: Determine the Degrees of the Numerator and Denominator

The given function is: \[ g(x) = \frac{x^3 - 5x^2 + x - 1}{x^2 - 15} \]

The degree of the numerator \(x^3 - 5x^2 + x - 1\) is 3, and the degree of the denominator \(x^2 - 15\) is 2.

Step 2: Compare the Degrees

To find the horizontal asymptote of a rational function, we compare the degrees of the numerator and the denominator:

If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\).
If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.

In this case, the degree of the numerator (3) is greater than the degree of the denominator (2).