Stitz-Zeager_College_Algebra_e-book

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Unformatted text preview: y = h(x) is a little bit below the line y = 2x − 1 as x → −∞. 256 Rational Functions 3 • The behavior of y = h(x) as x → ∞: If x → ∞, then x+2 ≈ very small (+). This means h(x) ≈ 2x − 1 + very small (+), or that the graph of y = h(x) is a little bit above the line y = 2x − 1 as x → ∞. This is end behavior unlike any we’ve ever seen. Instead of approaching a horizontal line, the graph is approaching a slanted line. For this reason, y = 2x − 1 is called a slant asymptote13 of the graph of y = h(x). A slant asymptote will always arise when the degree of the numerator is exactly one more than the degree of the denominator, and there’s no way to determine exactly what it is without going through the long division. Graphically we have y 4 3 2 1 x −1 −2 −3 −4 6. To make our sign diagram, we place an ‘ ’ above x = −2 and x = −1 and a ‘0’ above x = − 1 . 2 On our four test intervals, we ﬁnd h(x) is (+) on (−2, −1) and − 1 , ∞ and h(x) is (−) on 2 1 (−∞, −2) and −1, − 2...
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