Lesson 3.15
Horizontal Asymptotes (Top < Bottom)
When the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is always .
Introduction
Horizontal asymptotes describe where the graph goes as . The rule depends on comparing the degree of the numerator vs. denominator. This lesson covers the simplest case: top degree < bottom degree.
Past Knowledge
Polynomial degree (Chapter 5) and the reciprocal parent function (Lesson 3.12).
Today's Goal
Identify the HA when the numerator's degree is smaller than the denominator's degree.
Future Success
Lessons 3.16 and 3.17 cover the "equal degree" and "top > bottom" cases.
Key Concepts
The Three HA Rules (Overview)
deg(top) < deg(bottom) →
← This lesson
deg(top) = deg(bottom) → ratio of leading coefficients
Lesson 3.16
deg(top) > deg(bottom) → no HA (slant asymptote possible)
Lesson 3.17
Why ?
Consider . As gets huge:
The denominator grows much faster than the numerator, driving the fraction to zero.
Worked Examples
Example 1: Degree 0 vs. Degree 1
BasicFind the HA of .
Top degree = 0, bottom degree = 1. Top < Bottom.
HA:
Example 2: Degree 1 vs. Degree 2
IntermediateFind the HA of .
Top degree = 1, bottom degree = 2. Top < Bottom.
HA: (the graph CAN cross the HA here — and it does at the origin!)
Example 3: Degree 0 vs. Degree 2
AdvancedFind the HA of .
Top degree = 0, bottom degree = 2. Top < Bottom.
HA: , VAs:
Common Pitfalls
Thinking the HA Can't Be Crossed
Unlike vertical asymptotes, the graph can cross a horizontal asymptote! The HA only describes behavior as .
Comparing After Cancelling
Always compare degrees of the original numerator and denominator, not the simplified version.
Real-Life Applications
In pharmacology, the concentration of a drug in the bloodstream often follows a model where the HA at represents the drug being fully metabolized over time — the concentration approaches zero as .
Practice Quiz
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