NPR1 Section 3.5 Limits at Infinity

NPR2 Discuss “end behavior” of a function on an interval Discuss “end behavior” of a function on an interval Graph: Graph:

NPR3 As x increases without bound f(x) approaches _______. As x increases without bound f(x) approaches _______. As x decreases without bound f(x) approaches _______. As x decreases without bound f(x) approaches _______.

NPR4 Definition of a Horizontal Asymptote The line y=L is a horizontal asymptote The line y=L is a horizontal asymptote of the graph of f if of the graph of f if Note: a function can have at most 2 horizontal asymptotes

NPR5 Exploration Use a graphing utility to graph Use a graphing utility to graph y=(2x^2 +4x-6)/(3x^2+2x-16) y=(2x^2 +4x-6)/(3x^2+2x-16) Describe all important features of the graph. Describe all important features of the graph. Can you find a single viewing window that shows all these features clearly? Can you find a single viewing window that shows all these features clearly? What are the horizontal asymptotes? What are the horizontal asymptotes?

NPR6

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 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.

NPR8 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 greater than the degree of the denominator, then the limit of the rational function DNE.

NPR9 Examples 1) 1) 2) 2) 3) 3) 4) 4)

NPR10 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. 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: EX:

NPR11 The graph:

NPR12 Limits involving Trig Functions 1)2)

NPR13 Infinite Limits at Infinity Examples:1)3)2)4)

NPR14 Asymptotes 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. 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.

NPR15 Find an equation for the slant asymptote: Use long Division: Use long Division:

NPR16 References Larson, Hostetler, and Edwards, Calculus of a Single Variable, Houghton Mifflin Company: 2002 Larson, Hostetler, and Edwards, Calculus of a Single Variable, Houghton Mifflin Company: 2002