Asymptotes of Rational Functions

In AP Precalculus, asymptotes describe how a rational function behaves near values that it cannot reach or as the input becomes very large or very small.
Asymptotes are essential for understanding:

• Function behavior and limits

• Graph sketching

• End behavior and discontinuities

• Preparation for calculus concepts

A rational function may have vertical, horizontal, or slant (oblique) asymptotes.

Review: Rational Functions

A rational function has the form:

$$f(x)=\frac{p(x)}{q(x)}$$​

Where p(x) and q(x) are polynomials and q(x)≠0.

Vertical Asymptotes

Definition

A vertical asymptote occurs at x=a if:

$$\lim_{x\toa^{\pm}}f(x)=\pm$$

This happens when the denominator is zero after simplification.

How to Find Vertical Asymptotes

1. Factor numerator and denominator

2. Cancel common factors

3. Set remaining denominator factors equal to zero

Example

$$f(x)=\frac{x^2-1}{x^2-4}$$​

Factoring:

$$\frac{(x+1)(x-1)}{(x+2)(x-2)}$$​

Vertical asymptotes:

$$x=\pm2$$

Holes vs. Vertical Asymptotes

If a factor cancels, it creates a hole, not an asymptote.

Example:

$$\frac{x-3}{x-3} \longrightarrow$$ hole at x=3

Horizontal Asymptotes

Definition

A horizontal asymptote describes the end behavior of a rational function:

$$\lim_{x\to\pm\infty}f(x)$$

Rules for Horizontal Asymptotes

Compare degrees of numerator and denominator.

Case 1: Degree of numerator < degree of denominator

$$f(x)=\frac{2x+1}{x^2+5}$$​

Horizontal asymptote:

Case 2: Degrees are equal

$$f(x)=\frac{3x^2+1}{2x^2-7}$$​

Horizontal asymptote:

Case 3: Degree of numerator > degree of denominator

• No horizontal asymptote

• May have a slant asymptote

Slant (Oblique) Asymptotes

Definition

A slant asymptote occurs when:

$$p(x)=$$

How to Find a Slant Asymptote

Perform polynomial division.

Example

$$f(x)=\frac{x^2+3x+1}{x+1}$$​

Divide:

$$f(x)=x+2-\frac{1}{x+1}$$​

Slant asymptote:

End Behavior and Asymptotes

As $$x\longrightarrow \pm\infty$$, rational functions approach:

• A horizontal asymptote

• Or a slant asymptote

Asymptotes describe long-term behavior, not exact values.

Graphing with Asymptotes

To sketch a rational function:

1. Find vertical asymptotes

2. Identify holes

3. Determine horizontal or slant asymptotes

4. Plot zeros and test points

Asymptotes guide the overall shape of the graph.

Behavior Near Vertical Asymptotes

Evaluate one-sided limits:

$$\lim_{x\to a^{-}}f(x)$$ , $$\lim_{x\to a^{+}}f(x)$$

This determines whether the function approaches $$\infty$$ or $$-\infty$$.

Common Errors

• Forgetting to simplify before finding asymptotes

• Confusing holes with vertical asymptotes

• Assuming graphs touch asymptotes

• Incorrect degree comparisons

Summary

• Vertical asymptotes occur at non-canceled denominator zeros

• Horizontal asymptotes depend on degree comparison

• Slant asymptotes require polynomial division

• Asymptotes describe function behavior, not exact values

• Mastery of asymptotes is essential for AP Precalculus and Calculus