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Unformatted text preview: is no way any environment can support inﬁnitely
many bacteria, so shortly before t = 5 the environment would collapse.
Now that we have thoroughly investigated vertical asymptotes, we now turn our attention to
horizontal asymptotes. The next theorem tells us when to expect horizontal asymptotes.
Theorem 4.2. Location of Horizontal Asymptotes: Suppose r is a rational function and
r(x) = p(x) , where p and q are polynomial functions with leading coeﬃcients a and b, respectively.
• If the degree of p(x) is the same as the degree of q (x), then y =
of the graph of y = r(x). a
b is thea horizontal asymptote • If the degree of p(x) is less than the degree of q (x), then y = 0 is the horizontal asymptote
of the graph of y = r(x).
• If the degree of p(x) is greater than the degree of q (x), then the graph of y = r(x) has no
a The use of the deﬁnite article will be justiﬁed momentarily. 4.1 Introduction to Rational Functions 239 Like Theorem 4.1, Theorem 4.2 is proved using Calculus. Nevertheless, we can understand the idea
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