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1 Graph Sketching: Asymptotes and Rational Functions OBJECTIVES Find limits involving infinity. Determine the asymptotes of a function’s graph.

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2 A rational function is a function f that can be described by where P(x) and Q(x) are polynomials, with Q(x) not the zero polynomial. The domain of f consists of all inputs x for which Q(x) ≠ 0. DEFINITION:

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3 ASYMPTOTES A vertical asymptote is a vertical line that a function approaches, but never reaches. If a is a value for x that makes the denominator = 0, the line x = a is a vertical asymptote. The graph will never cross a vertical asymptote

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4 The line x = a is a vertical asymptote if any of the following limit statements are true: x = a ∞ ∞ -∞

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5 ASYMPTOTES A horizontal asymptote behaves differently. We determine what happens to the function as x approaches positive or negative infinity. The graph of the function can cross the horizontal asymptote. y = 0

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6 EXAMPLE 1: Graph the function Does not exist x-intercepty-intercept does not exist There is a vertical asymptote at x = 1

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7 EXAMPLE 1: Graph the function Now lets look at the one-sided limits around x = 1 x y x y ∞-∞ 1 ∞

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8 EXAMPLE 1: Graph the function There is a horizontal asymptote at y = 0 x = 1 y = 0 (0, -3) ∞ -∞

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9 EXAMPLE 2: Graph the function x-intercepty-intercept

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10 EXAMPLE 2: Graph the function does not exist There is a vertical asymptote at x = 3 x = 3

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11 Now lets look at the one-sided limits around x = 3 ∞ -∞

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12 Look for Horizontal Asymptotes There is a horizontal asymptote at y = 2 x = 3 y = 2 -∞ ∞

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13 EXAMPLE 2: Graph the function x = 3 y = 2 -∞ ∞

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14 EXAMPLE 3: Graph the function x- intercepty-intercept Domain: x ≠ -1, 3

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15 EXAMPLE 3: cont There is a 'hole' at (-1, ¾ )

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16 EXAMPLE 3: cont DNE There is a vertical asymptote at x = 3 x = 3

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17 f(3.001) = 1001 ∞ f(2.999) = ∞ Look at one-sided limits around x = 3 From the left From the right x = 3 -∞ ∞

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18 Is there a horizontal asymptote? There is a horizontal asymptote at y = 1 x = 3 -∞ ∞ y = 1

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19 (2, 0) y = 1 x = 3 ∞ - ∞

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