1. Home
  2. >Calculus 1
  3. >Limits
Help

Limits at infinity - horizontal asymptotes

0/4
?
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
0/12
?
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