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Chapter 3: Applications of Differentiation L3.5 Limits at Infinity

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Limits at Infinity End behavior of a function over an infinite interval left end behavior right end behavior If a function grows without bound or oscillates as x→ − ∞ or x→∞, then dne If the line y = L is a horizontal asymptote of the graph of f, then or The above implies that a function can have up to two horizontal asymptotes (HAs) –Rational functions have at most one HA –Functions that are not rational can have up to two HAs

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Limits at Infinity: Rational Functions Rational Functions have up to one horizontal asymptote: –HA is y = 0 if deg(numerator) < deg(denominator) –HA is y = ‘ratio of leading coefficients’ if deg(num) = deg(denom) –Otherwise, no horizontal asymptote Examples: 1. a. b. 2. 3. 4.

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Limits at Infinity: Non Rational Functions Functions that are not rational (include ) may approach different horizontal asymptotes at the left and right ends. Example: For x > 0,. So For x < 0,. So Recall that

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9.3 Rational Functions and Their Graphs

9.3 Rational Functions and Their Graphs

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