EXAMPLE 17
D. LIMIT OF A QUOTIENT OF POLYNOMIALS
To find where P(x) and Q(x) are polynomials in x, we can divide both numerator and
denominator by the highest power of x that occurs and use the fact that
EXAMPLE 18EXAMPLE 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 (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