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+1002x28lim_{x \to \infty } \;\frac{{-5x^2+13x+100}}{{2x^2-8}}
          (Type 1: degree of numerator = degree of denominator)
        2. limx  2x9x3+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  x23x+115xlim_{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+7x10005x+8f\left( x \right) = \frac{{\sqrt {3{x^2} + 7x - 1000} }}{{5x + 8}} by evaluating:
        i) limx  3x2+7x10005x+8lim_{x \to \infty } \;\frac{{\sqrt {3{x^2} + 7x - 1000} }}{{5x + 8}}
        ii) limx  3x2+7x10005x+8lim_{x \to - \infty } \;\frac{{\sqrt {3{x^2} + 7x - 1000} }}{{5x + 8}}
        1. Multiply Conjugates First, then Evaluate Limits
          Find:
          1. limx  (9x2+12x3x)lim_{x \to \infty } \;\left( {\sqrt {9{x^2} + 12x} - 3x} \right)
          2. limx  (x+x25x)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  x2xlim_{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  sinxlim_{x \to \infty } \;\sin x