###### Example 19 **__**

**__**###### Example 20 **__**

**__**###### The Rational Function Theorem

We see from Examples 18, 19, and 20 that: if the degree of P(x) is less than that

of Q(x), then ; if the degree of P(x) is higher than that of Q(x), then

` or −∞ (i.e., does not exist); and if the degrees of P(x) and Q(x) are the`

same, then , where an and bn are the coefficients of the highest

powers of x in P(x) and Q(x), respectively.

This theorem holds also when we replace “x→∞” by “x→−∞.”

Note also that:

(i) when , then y = 0 is a horizontal asymptote of the graph of

;

(ii) when or −∞, then the graph of has no horizontal

asymptotes;

(iii) when , then is a horizontal asymptote of the graph of

.

###### Example 21 **__**

**__**