Lesson 3.17

Horizontal Asymptotes (Top > Bottom)

When the degree of the numerator exceeds the denominator, there is no horizontal asymptote. Instead, the graph grows without bound — but when the numerator is exactly one degree higher, there's a slant (oblique) asymptote.

Introduction

This is the final HA case. When the numerator "wins" the degree comparison, the values blow up — no HA. But there's a special sub-case: if the numerator's degree is exactly one more than the denominator's, we get a slant asymptote found via polynomial long division.

Past Knowledge

Polynomial long division (Lesson 2.7), synthetic division (2.8), and the other HA rules (3.15-3.16).

Today's Goal

Determine when there's no HA, and find slant asymptotes using long division.

Future Success

The complete asymptote toolkit is essential for sketching rational graphs in Lesson 3.18.

Key Concepts

Complete HA Rules

deg(top) < deg(bottom) →

deg(top) = deg(bottom) → ratio of leading coefficients

deg(top) > deg(bottom) → NO HA

If exactly 1 degree higher → slant asymptote ← This lesson

Finding a Slant Asymptote

When: deg(top) = deg(bottom) + 1

Divide the numerator by the denominator using long division. The quotient (ignoring the remainder) is the slant asymptote.

Worked Examples

Example 1: Degree 2 over Degree 1

Basic

Find the slant asymptote of .

Factor numerator

The simplified form IS the slant asymptote

Since it cancels completely, the graph is just the line with a hole.

Slant Asymptote: (hole at )

Example 2: Long Division Required

Intermediate

Find the slant asymptote of .

Divide

The quotient is the slant asymptote (remainder goes to 0)

Slant Asymptote:

Example 3: Degree Difference > 1

Advanced

Does have a slant asymptote?

Degree difference = 3 − 1 = 2 (not exactly 1)

No HA and no slant asymptote. The end behavior is parabolic.

No HA, no slant asymptote — end behavior resembles

Common Pitfalls

Including the Remainder

The slant asymptote is only the quotient. Drop the remainder — it goes to zero as .

Saying "Slant" When Degree Diff ≠ 1

A slant asymptote only exists when the degree difference is exactly 1. Otherwise, no slant.

Real-Life Applications

In economics, average cost functions like often produce slant asymptotes that represent the long-run marginal cost. The oblique line shows where costs stabilize as production scales.