Rational Functions and Limits- Complete Guide

What Are Rational Functions?

A rational function is simply one polynomial divided by another polynomial. That's it. Nothing fancy.

The basic form looks like this:

R(x) = P(x) / Q(x)

Where P(x) and Q(x) are polynomials, and Q(x) ≠ 0. If Q(x) = 0, you've got problems — those x-values aren't in the domain.

Examples:

• f(x) = (x² + 3x - 4) / (x - 2)
• g(x) = 1/x
• h(x) = (2x³ - 5x + 1) / (x² - 4)

The Domain Problem

Finding the domain is straightforward. Exclude any x-value that makes the denominator zero.

Take f(x) = 1/(x-3). The denominator is zero when x = 3. So the domain is all real numbers except x = 3.

When you factor numerator and denominator, you can spot holes and vertical asymptotes before you ever touch a graph.

Asymptotes: The Skeleton of Rational Functions

Asymptotes tell you where the function goes as it approaches certain x-values or infinity. There are three types.

Vertical Asymptotes

These appear at x-values where the denominator equals zero and the numerator doesn't also equal zero at that point.

Rule: If (x - a) is a factor of the denominator but not a factor of the numerator, then x = a is a vertical asymptote.

Example: f(x) = 1/(x-2). Vertical asymptote at x = 2.

Horizontal Asymptotes

Horizontal asymptotes describe end behavior — what happens as x approaches infinity or negative infinity.

How to find them:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
• If the degrees are equal, the horizontal asymptote is y = (leading coefficient of numerator) / (leading coefficient of denominator).
• If the numerator has the higher degree, there is no horizontal asymptote. You might have an oblique asymptote instead.

Oblique (Slant) Asymptotes

When the numerator's degree is exactly one higher than the denominator's degree, you get a slant asymptote.

Find it by polynomial long division. The quotient (ignoring the remainder) is your oblique asymptote.

Holes in the Graph

A hole appears when the same factor exists in both numerator and denominator.

Example: f(x) = (x² - 4)/(x - 2)

Factor the numerator: (x - 2)(x + 2)/(x - 2). Cancel (x - 2) and you get x + 2.

But that canceled factor means there's a hole at x = 2. Plug x = 2 into x + 2 and you get 4. So the point (2, 4) is missing from the graph.

Limits and Rational Functions

Limits matter when direct substitution gives you 0/0 — the indeterminate form.

When you see 0/0, don't panic. Factor and cancel. Then try substitution again.

Example: Find lim(x→2) of (x² - 4)/(x - 2)

Direct substitution: (4 - 4)/(2 - 2) = 0/0. Useless.

Factor: (x - 2)(x + 2)/(x - 2) = x + 2

Now substitute: 2 + 2 = 4

The limit exists and equals 4.

Limits at Infinity

As x → ∞, rational functions behave predictably:

• Compare degrees of numerator and denominator
• Degree numerator < degree denominator → limit is 0
• Degrees equal → limit is the ratio of leading coefficients
• Degree numerator > degree denominator → limit is ±∞ or doesn't exist

Comparing Asymptote Types

Graphing Rational Functions: Getting Started

Here's the actual process:

Step 1: Find the Domain

Set denominator ≠ 0. Write down all excluded x-values.

Step 2: Simplify First

Factor numerator and denominator. Cancel any common factors. Note any holes.

Step 3: Find Vertical Asymptotes

At each x-value excluded from the domain, check if it's a hole or a vertical asymptote. Holes come from canceled factors. Vertical asymptotes come from factors that remain in the denominator.

Step 4: Find Horizontal or Oblique Asymptotes

Compare degrees. Use the rules from the table above.

Step 5: Find Y-Intercept

Plug x = 0 into the simplified function. If 0 is in the domain.

Step 6: Find X-Intercepts

Set the numerator equal to zero (in the simplified function). These are your x-intercepts.

Step 7: Plot Key Points and Draw

Sketch the asymptotes. Plot holes, intercepts. Determine sign between critical points to know where the graph sits above or below the x-axis.

Common Mistakes to Avoid

• Forgetting to factor first. You can't find holes or simplify without factoring.
• Canceling before factoring. Factor, then cancel. Not the other way around.
• Ignoring holes. A hole is a point missing from the graph. It affects the shape.
• Confusing vertical asymptotes with slant asymptotes. Vertical comes from denominator factors. Slant comes from division.
• Using direct substitution on 0/0. Always factor first.

Quick Reference: What to Do When