Question

How do you find the vertical and horizontal asymptotes $?$

Answer 1

Find the vertical asymptotes of a rational function:

A rational function $R\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$ in lowest terms will have a vertical asymptote $x=r$ if $r$ is a real zero of the denominator $q\left(x\right)$. That is, if $x-r$ is a factor of the denominator $q\left(x\right)$ of a rational function $R\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$, in lowest terms, $R\left(x\right)$ will have the vertical asymptote $x=r$.

Ways to find the horizontal asymptote of a rational function

Consider the rational function $R\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}=\frac{a_n{x}^n+a_{n-1}{x}^{n-1}+....+a_{1}x+a_0}{b_m{x}^m+b_{m-1}{x}^{m-1}+....+b_{1}x+b_0}$ in which the degree of the numerator is $n$ and the degree of the denominator is $m$.
1. If $n<m$ (the degree of the numerator is less than the degree of the denominator), the line $y=0$ is a horizontal asymptote.
2. If $n=m$(the degree of the numerator equals the degree of the denominator), the line $y=\frac{a_n}{b_m}$ is a horizontal asymptote. (that is, the horizontal asymptote equals the ratio of the leading coefficients.)
3. If $n=m+1$ (the degree of the numerator is one more than the degree of the denominator), then there is no horizontal asymptote. If $R$ in lowest terms is a linear
function, then we agree that $R$ has no horizontal asymptote.
4. If $n\ge m+2$ (the degree of the numerator is two or more greater than the degree of the denominator), there are no horizontal or oblique asymptotes.

For eg. $R\left(x\right)=\frac{x^2-4}{x-3}$

Vertical asymptotes of $R$

$R$ is in lowest terms, and the zero of the denominator $x$ is 3. The lines $x=3$ is the vertical asymptote of the graph of $R$.

Horizontal Asymptotes of $R$

Since the degree of numerator is one more than the degree of the denominator, there is no horizontal asymptote.

Hence, the lines $x=3$ is the vertical asymptote of the graph of $R$ and there is no horizontal asymptote of $R$.