Vertical Asymptotes -- Basic Rules

A rational function **f(x) = p(x) / q(x)** may have a vertical asymptote **x=a** only if the denominator, **q(x)**, is zero at **x=a** .

If the denominator is 0 at **x=a**, and the numerator is different from 0 at **x=a**, then **f(x)** does have a vertical asymptote at **x=a** .

Horizontal Asymptotes -- Basic Rules

Consider a rational function f(x) = p(x) / q(x)

1. If the degree of p(x) is equal to the degree of q(x), then

** y=a_n / b_n**

is a horizontal asymptote, where **a_n** and **b_n** are leading coefficients of **p(x)** and **q(x)**, respectively.

2. If the degree of the denominator, **q(x)**, is larger than the degree of the numerator, **p(x)**, then **y=0** is a horizontal asymptote.

3. If the degree of the denominator, **q(x)**, is less than the degree of the numerator, **p(x)**, then **f(x)** has no horizontal asymptote. (It may have a slant asymptote.)