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 thesame, 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
.