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Horizontal asymptote - Rational Functions

Horizontal asymptote

Lessons

Notes:

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

  • 2.
    Algebraic Analysis on Horizontal Asymptotes

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

  • 3.
    Graphing Rational Functions

    Sketch each rational function by determining:

    i) vertical asymptote.

    ii) horizontal asymptotes

  • 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

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Horizontal asymptote

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