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Limits at infinity - horizontal asymptotes

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Intros
Lessons
  1. Introduction to Horizontal Asymptotes
  2. opposite relationship between "vertical asymptote" and "horizontal asymptote"
  3. how horizontal asymptotes are defined on each end of a function
  4. evaluate limits at infinity algebraically –"Highest Power Rule"!
  5. lesson overview
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Examples
Lessons
  1. Relate Asymptotes to Limits
    Express all asymptotes in limit notations for the function f f whose graph is shown below.
    Finding limits of a function algebraically by direct substitution
    1. Discuss the Foundation of Limits at Infinity
      Find:
      i) limx→∞  1xlim_{x \to \infty } \;\frac{1}{x}
      ii) limx→−∞  1xlim_{x \to - \infty } \;\frac{1}{x}
      1. Use "Highest Power Rule" to Evaluate Limits at Infinity of Rational Functions in 3 Types
        Find:
        1. limx→∞  −5x2+13x+1002x2−8lim_{x \to \infty } \;\frac{{-5x^2+13x+100}}{{2x^2-8}}
          (Type 1: degree of numerator = degree of denominator)
        2. limx→−∞  2x−9x3+7x2+10x+21lim_{x \to - \infty } \;\frac{{2x - 9}}{{{x^3} + 7{x^2} + 10x + 21}}
          (Type 2: degree of numerator < degree of denominator)
        3. limx→∞  x2−3x+115−xlim_{x \to \infty } \;\frac{{{x^2} - 3x + 11}}{{5 - x}}
          (Type 3: degree of numerator > degree of denominator)
      2. Evaluate Limits at Infinity of Functions Involving Radicals
        Find the horizontal asymptotes of the function f(x)=3x2+7x−10005x+8f\left( x \right) = \frac{{\sqrt {3{x^2} + 7x - 1000} }}{{5x + 8}} by evaluating:
        i) limx→∞  3x2+7x−10005x+8lim_{x \to \infty } \;\frac{{\sqrt {3{x^2} + 7x - 1000} }}{{5x + 8}}
        ii) limx→−∞  3x2+7x−10005x+8lim_{x \to - \infty } \;\frac{{\sqrt {3{x^2} + 7x - 1000} }}{{5x + 8}}
        1. Multiply Conjugates First, then Evaluate Limits
          Find:
          1. limx→∞  (9x2+12x−3x)lim_{x \to \infty } \;\left( {\sqrt {9{x^2} + 12x} - 3x} \right)
          2. limx→−∞  (x+x2−5x)lim_{x \to - \infty } \;\left( {x + \sqrt {{x^2} - 5x} } \right)
        2. Infinite Limits at Infinity
          Find:
          i) limx→∞  x3lim_{x \to \infty } \;{x^3}
          ii) limx→−∞  x3lim_{x \to - \infty } \;{x^3}
          1. Ambiguous Case: ∞−∞ \infty - \infty
            Find limx→∞  x2−xlim_{x \to \infty } \;{x^2} - x
            1. Limits at Infinity of Exponential Functions
              Find:
              i) limx→∞  exlim_{x \to \infty } \;{e^x}
              ii) limx→−∞  exlim_{x \to - \infty } \;{e^x}
              1. Limits at Infinity of Trigonometric Functions
                Find limx→∞  sin⁡xlim_{x \to \infty } \;\sin x
                Topic Notes
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                There are times when we want to see how a function behaves near a horizontal asymptote. Much like finding the limit of a function as x approaches a value, we can find the limit of a function as x approaches positive or negative infinity. In this section, we will learn how to evaluate limits at infinity algebraically using the "Highest Power Rule", with tricks like using conjugates, common denominators, and factoring.
                The line y=Ly = L is a horizontal asymptote of the curve y=f(x)y = f(x) in any of the following two cases:
                i)limx→∞f(x)=Llim_{x \to\infty } f\left( x \right) = L
                ii) limx→,−∞f(x)=Llim_{x \to,-\infty } f\left( x \right) = L

                horizontal asymptote in limit notation positive infinity

                horizontal asymptote in limit notation negative infinity

                or

                horizontal asymptote in limit notation positive infinity 2

                or

                horizontal asymptote in limit notation negative infinity 2