Unformatted text preview: about end behavior of polynomials to help us understand
this theorem. From Theorem 3.2, we know the end behavior of a polynomial is determined by its
leading term. Applying this to the numerator and denominator of f (x), we get that as x → ±∞,
f (x) = 2x+11 ≈ 2x = 2. This last approach is useful in Calculus, and, indeed, is made rigorous there.
(Keep this in mind for the remainder of this paragraph.) Applying this reasoning to the general
case, suppose r(x) = p(x) where a is the leading coeﬃcient of p(x) and b is the leading coeﬃcient
of q (x). As x → ±∞, r(x) ≈ bxm , where n and m are the degrees of p(x) and q (x), respectively.
If the degree of p(x) and the degree of q (x) are the same, then n = m so that r(x) ≈ a , which
means y = a is the horizontal asymptote in this case. If the degree of p(x) is less than the degree
of q (x), then n < m, so m − n is a positive number, and hence, r(x) ≈ bxm−n → 0 as x → ±∞. If
the degree of p(x) is greater than the degree of q (x), then n > m, and hen...
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