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 → ±∞,
x−
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
(x)
case, suppose r(x) = p(x) where a is the leading coeﬃcient of p(x) and b is the leading coeﬃcient
q
n ax
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
b
means y = a is the horizontal asymptote in this case. If the degree of p(x) is less than the degree
b
a
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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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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