Horizontal asymptote

Horizontal asymptote

Lessons

There are 3 cases to consider when determining horizontal asymptotes:

1) Case 1:

if: degree of numerator < degree of denominator

then: horizontal asymptote: y = 0 (x-axis)

i.e.f(x)=ax3+......bx5+......i.e. f(x) = \frac{ax^{3}+......}{bx^{5}+......} → horizontal asymptote: y=0y = 0

2) Case 2:

if: degree of numerator = degree of denominator

then: horizontal asymptote: y = leadingcoefficientofnumeratorleadingcoefficientofdenominator\frac{leading\; coefficient \;of\; numerator}{leading\; coefficient\; of\; denominator}

i.e.f(x)=ax5+......bx5+......i.e. f(x) = \frac{ax^{5}+......}{bx^{5}+......} → horizontal asymptote: y=aby = \frac{a}{b}

3) Case 3:

if: degree of numerator > degree of denominator

then: horizontal asymptote: NONE

i.e.f(x)=ax5+......bx3+......i.e. f(x) = \frac{ax^{5}+......}{bx^{3}+......} NOhorizontalasymptote NO\; horizontal\; asymptote

  • 1.
    Introduction to Horizontal Asymptote

    • Horizontal Asymptotes define the right-end and left-end behaviors on the graph of a function.

    • 3 cases of horizontal asymptotes in a nutshell…

  • 2.
    Algebraic Analysis on Horizontal Asymptotes

    Let’s take an in-depth look at the reasoning behind each case of horizontal asymptotes:

    a)
    Case 1:

    if: degree of numerator < degree of denominator

    then: horizontal asymptote: y = 0 (x-axis)

    i.e.f(x)=ax3+......bx5+......i.e. f(x) = \frac{ax^{3}+......}{bx^{5}+......} → horizontal asymptote: y=0y = 0


    b)
    Case 2:

    if: degree of numerator = degree of denominator

    then: horizontal asymptote: y = leadingcoefficientofnumeratorleadingcoefficientofdenominator\frac{leading\; coefficient \;of\; numerator}{leading\; coefficient\; of\; denominator}

    i.e.f(x)=ax5+......bx5+......i.e. f(x) = \frac{ax^{5}+......}{bx^{5}+......} → horizontal asymptote: y=aby = \frac{a}{b}


    c)
    Case 3:

    if: degree of numerator > degree of denominator

    then: horizontal asymptote: NONE

    i.e.f(x)=ax5+......bx3+......i.e. f(x) = \frac{ax^{5}+......}{bx^{3}+......} NOhorizontalasymptote NO\; horizontal\; asymptote



  • 3.
    Graphing Rational Functions

    Sketch each rational function by determining:

    i) vertical asymptote.

    ii) horizontal asymptotes

    a)
    f(x)=52x+10f\left( x \right) = \frac{5}{{2x + 10}}

    b)
    g(x)=5x213x+62x2+3x+2g\left( x \right) = \frac{{5{x^2} - 13x + 6}}{{ - 2{x^2} + 3x + 2}}

    c)
    h(x)=x320x100h\left( x \right) = \frac{{{x^3}}}{{20x - 100}}


  • 4.
    Identifying Characteristics of Rational Functions

    Without sketching the graph, determine the following features for each rational function:

    i) point of discontinuity

    ii) vertical asymptote

    iii) horizontal asymptote

    iv) slant asymptote

    a)
    a(x)=x9x+9a(x) = \frac{x - 9}{x + 9}

    b)
    b(x)=x29x2+9b(x) = \frac{x^{2}-9}{x^{2}+9}

    c)
    c(x)=x2+9x29c(x) = \frac{x^{2}+9}{x^{2}-9}

    d)
    d(x)=x+9x29d(x) = \frac{x+9}{x^{2}-9}

    e)
    e(x)=x+3x29e(x) = \frac{x+3}{x^{2}-9}

    f)
    f(x)=x2+9x+9f(x) = \frac{x^{2}+9}{x+9}

    g)
    g(x)=x9x29g(x) = \frac{-x-9}{-x^{2}-9}

    h)
    h(x)=x29x2+9h(x) = \frac{-x^{2}-9}{-x^{2}+9}

    i)
    i(x)=x29x+3i(x) = \frac{x^{2}-9}{x+3}

    j)
    j(x)=x39x2x23xj(x) = \frac{x^{3}-9x^{2}}{x^{2}-3x}