Horizontal asymptote
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- Algebraic Analysis on Horizontal Asymptotes
Let's take an in-depth look at the reasoning behind each case of horizontal asymptotes:
- Case 1:
if: degree of numerator < degree of denominator
then: horizontal asymptote: y = 0 (x-axis)
i.e.f(x)=bx5+......ax3+......→ horizontal asymptote: y=0
- Case 2:
if: degree of numerator = degree of denominator
then: horizontal asymptote: y = leadingcoefficientofdenominatorleadingcoefficientofnumerator
i.e.f(x)=bx5+......ax5+......→ horizontal asymptote: y=ba
- Case 3:
if: degree of numerator > degree of denominator
then: horizontal asymptote: NONE
i.e.f(x)=bx3+......ax5+......→NOhorizontalasymptote
- Case 1:
- Graphing Rational Functions
Sketch each rational function by determining:
i) vertical asymptote.
ii) horizontal asymptotes
- 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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Topic 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)=bx5+......ax3+......→ horizontal asymptote: y=0
2) Case 2:
if: degree of numerator = degree of denominator
then: horizontal asymptote: y = leadingcoefficientofdenominatorleadingcoefficientofnumerator
i.e.f(x)=bx5+......ax5+......→ horizontal asymptote: y=ba
3) Case 3:
if: degree of numerator > degree of denominator
then: horizontal asymptote: NONE
i.e.f(x)=bx3+......ax5+......→NOhorizontalasymptote
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