Question

which asymptote(s) are determined by looking at the denominator? vertical horizontal slant none

Explanation:

Step1: Recall the definition of vertical asymptote

For a rational function \(y = \frac{f(x)}{g(x)}\), vertical asymptotes occur at the values of \(x\) that make the denominator \(g(x)=0\) (provided the numerator \(f(x)
eq0\) at those \(x\) - values).

Step2: Analyze horizontal asymptote

The horizontal asymptote of a rational function \(y=\frac{f(x)}{g(x)}=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}\) is determined by comparing the degrees \(n\) (of the numerator) and \(m\) (of the denominator). If \(n < m\), \(y = 0\); if \(n=m\), \(y=\frac{a_n}{b_m}\); if \(n>m\), there is no horizontal asymptote. It is not directly determined by just looking at the denominator.

Step3: Analyze slant asymptote

A slant asymptote of a rational function \(y=\frac{f(x)}{g(x)}\) occurs when the degree of the numerator \(n\) is exactly one more than the degree of the denominator \(m\) (\(n=m + 1\)). We find the slant asymptote by performing polynomial long - division of \(f(x)\) by \(g(x)\). It is not directly determined by just looking at the denominator.

So, the asymptote determined by looking at the denominator is the vertical asymptote.

Answer:

1. vertical