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Calculus 2.2

Calculus 2.2. -∞Limits involving Infinity∞. Horizontal Asymptote of y = f(x). Is the line y = b if either lim x→∞ f(x) = b OR lim x→-∞ f(x) = b Try #5. Horizontal Asymptote of y = sinx/x. Graph/table suggest y = 0 Sandwich Theorem: for x > 0

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Calculus 2.2

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  1. Calculus 2.2 -∞Limits involving Infinity∞

  2. Horizontal Asymptote of y = f(x) • Is the line y = b if either limx→∞f(x) = b OR limx→-∞f(x) = b Try #5

  3. Horizontal Asymptote of y = sinx/x • Graph/table suggest y = 0 • Sandwich Theorem: for x > 0 -1/x≤ sinx/x ≤ 1/x and 0 = limx→∞ -1/x = limx→∞1/x Therefore, limx→∞sinx/x must also = 0. limx→-∞sinx/x = 0 because of symmetry of even functions(can you prove it’s even?) Try #10

  4. Properties of Limits as x→±∞ • See page 71! • SUM • DIFFERENCE • PRODUCT • CONSTANT MULTIPLE • QUOTIENT • POWER Try # 23 • Remember hypothesis: limits must exist before they can be combined arithmetically. See Exploration on page 72.

  5. Vertical Asymptote of y = f(x) • Is the line x = a if either limx→a+f(x) = ±∞ OR limx→a-f(x) = ±∞ • Try #17, 29

  6. End Behavior Models • Is a simple function g(x) which acts like a more complicated function f(x) at ∞ or at −∞ or both. • Right end behavior model: models behavior at ∞, the limitx→∞ f(x)/g(x) = 1 • Left end behavior model: models behavior at −∞, the limitx→−∞ f(x)/g(x) = 1

  7. 2.2 continued… • If single function models both behaviors, it is called an “end behavior model.” • Polynomial functions behave like their leading term….see ex 7 page 74 • Rational functions have simple power functions for end behavior models.

  8. Examples… • Try 35- 44

  9. Example 9 with “seeing limits.” • Try # 49

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