Lesson 6.6

Slant Asymptotes

When the numerator is exactly "one degree stronger" than the denominator, the graph doesn't level off—it shoots away in a straight diagonal line.

Introduction

If the top degree is exactly one greater (e.g., or ), the function will grow unbounded, but it will hug a tilted diagonal line called a Slant (or Oblique) Asymptote.

Past Knowledge

In Lesson 5.1, we mastered Long Division. In Lesson 6.4, we learned that if the Top Degree > Bottom Degree, there is no Horizontal Asymptote.

Today's Goal

Learn to find the equation of a Slant (or Oblique) Asymptote using polynomial division.

Future Success

This concept bridges the gap between horizontal limits and full curve sketching, setting the stage for graphing any rational function.

Key Concepts

The Division Strategy

To find the equation of the Slant Asymptote, simply perform the division.

The "Ignore the Remainder" Rule

As , the fraction shrinks to zero.
So, the graph behaves exactly like the Quotient part.
.

Worked Examples

Level: Basic

Example 1: Using Synthetic Division

Find the slant asymptote of .

Degrees check: Top = 2, Bottom = 1. Difference is 1. Yes, it has a Slant Asymptote.
-1
1
4
-5
↓
-1
-3
1
3
-8 (R)
Quotient: . Remainder: -8.
Answer
Asymptote Equation:
We discard the remainder (-8).
Level: Intermediate

Example 2: Complex Divisor

Find the SA for .

Degrees: Top 3, Bottom 2. The difference is 1, so yes, SA exists.
Divisor is (not linear), so we must use Long Division.
Quotient only concerns the first two terms:

Quick Trick: You can stop dividing once you reach the constant term!
Answer
Level: Advanced

Example 3: Checking the Rule

Does have a Slant Asymptote?

Top Degree: 4
Bottom Degree: 2
Difference = .
Since the difference is NOT 1, it is not line-like (linear).
It is actually a (Parabolic) Asymptote, but that is generally outside the scope of this course.
Answer
No Slant Asymptote.

Common Pitfalls

  • Including the Remainder:

    The equation should just be . Do not write . The whole point is that the fractional part goes away at infinity.

  • Using Synthetic Division incorrectly:

    Remember Synthetic Division only works easily for divisors like . If you have or , you might be safer sticking to Long Division to avoid errors.

Real-Life Applications

Economics: Average Cost

The cost to produce items is often .

The Average Unit Cost is .

As gets huge, the term vanishes (fixed costs are spread out), and your per-unit cost approaches the line .

Practice Quiz

Loading...