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Section 3.5 Limits at Infinity

Section 3.5 Limits at Infinity. Discuss “end behavior” of a function on an interval Graph:. As x increases without bound f(x) approaches _______. As x decreases without bound f(x) approaches _______. Definition of a Horizontal Asymptote. The line y=L is a horizontal asymptote

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Section 3.5 Limits at Infinity

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  1. Section 3.5Limits at Infinity NPR

  2. Discuss “end behavior” of a function on an interval • Graph: NPR

  3. As x increases without bound f(x) approaches _______. • As x decreases without bound f(x) approaches _______. NPR

  4. Definition of a Horizontal Asymptote • The line y=L is a horizontal asymptote of the graph of f if Note: a function can have at most 2 horizontal asymptotes NPR

  5. Exploration • Use a graphing utility to graph y=(2x^2 +4x-6)/(3x^2+2x-16) • Describe all important features of the graph. • Can you find a single viewing window that shows all these features clearly? • What are the horizontal asymptotes? NPR

  6. NPR

  7. Recall Limits at infinity: 1. If r is a positive rational number and c is any real number, then Furthermore, if x^r is defined when x<0, then 2. If the degree of the numerator is less than the degree of the denominator, then the limit of the rational function is 0. NPR

  8. 3. If the degree of the numerator is equal the degree of the denominator, then the limit of the rational function is the ratio of the leading coefficients. 4. If the degree of the numerator is greaterthan the degree of the denominator, then the limit of the rational function DNE. NPR

  9. Examples 1) 2) 3) 4) NPR

  10. More on Horizontal Asymptotes • Rational functions always have the same horizontal asymptote to the right and to the left. Functions that are NOT rational may approach different horizontal asymptotes. • EX: NPR

  11. The graph: NPR

  12. Limits involving Trig Functions 1) 2) NPR

  13. Examples: 1) 3) 2) 4) Infinite Limits at Infinity NPR

  14. Asymptotes • www.purplemath.com/modules/asymtote4.htm • Slant asymptotes: The graph of a rational function (having no common factors) has a slant asymptote if the degree of the numerator exceeds the degree of the denominator by 1. NPR

  15. Find an equation for the slant asymptote: • Use long Division: NPR

  16. References • Larson, Hostetler, and Edwards, Calculus of a Single Variable, Houghton Mifflin Company: 2002 NPR

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