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# Asymptotes and holes 97

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• 1. POINTS OF DISCONTINUITY (Holes and Asymptotes) Points of discontinuity occur when the denominator of a function equals zero. 2 2 2 2 1. 2 3 2. 4 3 1 3. 2 x x y x x y x x x y x x 1 1 2 x y x x 3 3 1 x y x x 2 1 2 x x y x
• 2. Horizontal Asymptotes Note: The graph of a rational function has at most one horizontal asymptote. If the degree of the numerator is less than the denominator, the horizontal asymptote is y = 0. If the degree of the numerator is more than the denominator, there is NO horizontal asymptote . If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is y = a / b , where a and b are the leading coefficients of the numerator and the denominator. 1 y x Example: 2 2 2 2 1 x x y x x Example: 2 1y xExample:
• 3. EXAMPLE: Finding the Slant/Oblique Asymptote of a Rational Function Find the slant asymptotes of f(x) 2 4 5 . 3 x x x Solution Because the degree of the numerator, 2, is exactly one more than the degree of the denominator, 1, the graph of f has a slant asymptote. To find the equation of the slant asymptote, divide x 3 into x2 4x 5: 2 1 4 5 1 3 3 1 1 8 3 2 8 1 1 3 3 4 5 x x x x x Remainder moremore
• 4. 1 2 1 1 x x y x x 2 2 2 1 x x y x Graph the equation.
• 5. 2 1 2 x y x x Graph the equation. 1 1 2 x y x x