Lesson 3.16

Horizontal Asymptotes (Top = Bottom)

When the numerator and denominator have the same degree, the horizontal asymptote is the ratio of the leading coefficients.

Introduction

In Lesson 3.15, the denominator "won" because it had a higher degree, pulling the function to zero. Now the degrees are tied, so neither wins — the function levels off at the ratio of lead coefficients.

Past Knowledge

Leading coefficients (Chapter 5) and HA rule for top < bottom (Lesson 3.15).

Today's Goal

Find the HA by dividing the leading coefficients when the degrees are equal.

Future Success

Lesson 3.17 covers the final case: top > bottom (no HA, slant asymptote instead).

Key Concepts

The Rule

If deg(numerator) = deg(denominator):

Example:

Why?

For huge , the lower-degree terms become negligible:

Only the highest-degree terms matter at infinity.

Worked Examples

Example 1: Both Degree 1

Basic

Find the HA of .

1

Both degree 1. Lead coefficients: 4 and 2.

HA:

Example 2: Both Degree 2

Intermediate

Find the HA of .

1

Both degree 2. Lead coefficients: 3 and 1.

HA: (no VA — denominator is always positive)

Example 3: Negative Ratio

Advanced

Find the HA of .

1

Both degree 3. Lead coefficients: −2 and 1.

HA:

Common Pitfalls

Using ALL Coefficients

Only use the leading coefficients (the ones attached to the highest power). The other coefficients don't affect the HA.

Forgetting the Negative

If the leading coefficient is negative, the HA is negative too. Don't drop the sign!

Real-Life Applications

In environmental science, population models with carrying capacity have an HA representing the maximum sustainable population. The ratio of leading coefficients determines that long-term limit.

Practice Quiz

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