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

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)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
  • Introduction
    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


  • 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

  • 2.
    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}

  • 3.
    Use "Highest Power Rule" to Evaluate Limits at Infinity of Rational Functions in 3 Types
    Find:
    a)
    limx  5x2+13x+1002x28lim_{x \to \infty } \;\frac{{-5x^2+13x+100}}{{2x^2-8}}
    (Type 1: degree of numerator = degree of denominator)

    b)
    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)

    c)
    limx  x23x+115xlim_{x \to \infty } \;\frac{{{x^2} - 3x + 11}}{{5 - x}}
    (Type 3: degree of numerator > degree of denominator)


  • 4.
    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}}

  • 5.
    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)


  • 6.
    Infinite Limits at Infinity
    Find:
    i) limx  x3lim_{x \to \infty } \;{x^3}
    ii) limx  x3lim_{x \to - \infty } \;{x^3}

  • 7.
    Ambiguous Case: \infty - \infty
    Find limx  x2xlim_{x \to \infty } \;{x^2} - x

  • 8.
    Limits at Infinity of Exponential Functions
    Find:
    i) limx  exlim_{x \to \infty } \;{e^x}
    ii) limx  exlim_{x \to - \infty } \;{e^x}

  • 9.
    Limits at Infinity of Trigonometric Functions
    Find limx  sinxlim_{x \to \infty } \;\sin x