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Published byMagdalen Potter Modified over 9 years ago
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Euclid was a Greek mathematician best known for his treatise on geometry: The Elements. This influenced the development of Western mathematics for more than 2000 years. Euclid was a Greek mathematician best known for his treatise on geometry: The Elements. This influenced the development of Western mathematics for more than 2000 years. Euclid of Alexandria 325 – 265 B.C. Euclid of Alexandria 325 – 265 B.C.
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So we say, This limit shows that f (x) has a horizontal asymptote at y = 1.
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We take limits to infinity to analyze end-behavior of functions.
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Left-Hand Behavior Right-Hand Behavior
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Example 1
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Example 2
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As the denominator gets larger, the value of the fraction gets closer to zero. So by definition, y = 0 is a horizontal asymptote. Recall vertical asymptote at x = 0. horizontal asymptote at y = 0. Example 3
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Find: x y When we graph this function, the limit appears to be zero. Example 4 (2)
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Thus, by the Squeeze Theorem: x y Example 4 When we graph this function, the limit appears to be zero. Find:
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Example 5 Evaluate the following limit
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Example 6 Dominating term
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Method IMethod II Algebraic Limits
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Dominating term effect (DTE) we get, dominant terms in numerator and denominator Example 7
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Recall: Example
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Example 8
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Example 9 Hence, y = 0 is a horizontal asymptote. a.
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Example 9 There is no left hand behavior because of the domain restriction. So, x = 0 is a vertical asymptote. Hence, there is no horizontal asymptote.
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Example 9
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Example 11 Example 10
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a. Find the domain of f (x). b. Determine any asymptotes for f (x). Example 12 Example 13 Determine if a steady state exists and give the steady-state value for the function below.
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