Limits at infinity - horizontal asymptotes

Limits at infinity - horizontal asymptotes

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.

Lessons

Express all asymptotes in limit notations for the function f whose graph is shown below.
i)limxf(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
  • 1.
    Introduction to Horizontal Asymptotes
    a)
    opposite relationship between “vertical asymptote” and “horizontal asymptote"

    b)
    how horizontal asymptotes are defined on each end of a function

    c)
    evaluate limits at infinity algebraically –“Highest Power Rule”!

    d)
    lesson overview


  • 2.
    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

  • 3.
    Discuss the Foundation of Limits at Infinity
    Find:
    i) limx1xlim_{x \to \infty } \;\frac{1}{x}
    ii) limx1xlim_{x \to - \infty } \;\frac{1}{x}

  • 4.
    Use "Highest Power Rule" to Evaluate Limits at Infinity of Rational Functions in 3 Types
    Find:
    a)
    limx5x2+13x+1002x28lim_{x \to \infty } \;\frac{{-5x^2+13x+100}}{{2x^2-8}}
    (Type 1: degree of numerator = degree of denominator)

    b)
    limx2x9x3+7x2+10x+21lim_{x \to - \infty } \;\frac{{2x - 9}}{{{x^3} + 7{x^2} + 10x + 21}}
    (Type 2: degree of numerator < degree of denominator)

    c)
    limxx23x+115xlim_{x \to \infty } \;\frac{{{x^2} - 3x + 11}}{{5 - x}}
    (Type 3: degree of numerator > degree of denominator)


  • 5.
    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) limx3x2+7x10005x+8lim_{x \to \infty } \;\frac{{\sqrt {3{x^2} + 7x - 1000} }}{{5x + 8}}
    ii) limx3x2+7x10005x+8lim_{x \to - \infty } \;\frac{{\sqrt {3{x^2} + 7x - 1000} }}{{5x + 8}}

  • 6.
    Multiply Conjugates First, then Evaluate Limits
    Find:
    a)
    limx(9x2+12x3x)lim_{x \to \infty } \;\left( {\sqrt {9{x^2} + 12x} - 3x} \right)

    b)
    limx(x+x25x)lim_{x \to - \infty } \;\left( {x + \sqrt {{x^2} - 5x} } \right)


  • 7.
    Infinite Limits at Infinity
    Find:
    i) limxx3lim_{x \to \infty } \;{x^3}
    ii) limxx3lim_{x \to - \infty } \;{x^3}

  • 8.
    Ambiguous Case: \infty - \infty
    Find limxx2xlim_{x \to \infty } \;{x^2} - x

  • 9.
    Limits at Infinity of Exponential Functions
    Find:
    i) limxexlim_{x \to \infty } \;{e^x}
    ii) limxexlim_{x \to - \infty } \;{e^x}

  • 10.
    Limits at Infinity of Trigonometric Functions
    Find limxsinxlim_{x \to \infty } \;\sin x