Topic 1.7A

Rational Functions and End Behavior

5 MIN READ7 IDEAS24 PROBLEMS14 flashcards

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01
"No horizontal asymptote" ≠ "no end behavior." The function still goes somewhere — it just isn't a flat line.

Practice 24 problems Next: Chapter 1.7B Rational Functions and End Behavior

What Is a Rational Function?

A rational function is a ratio of two polynomials — one polynomial divided by another:

f(x) = \frac{p(x)}{q(x)}, \qquad q(x) \neq 0

Just like you can never divide by zero in arithmetic, a rational function is undefined wherever its denominator equals zero. Those excluded x-values are simply left out of the domain.

f(x) = \frac{2x + 1}{x - 3} \qquad (x \neq 3)

CONCEPT

Example

For $f(x) = (2x+1)/(x-3)$ , the input $x = 3$ would make the denominator 0 — so $x = 3$ is excluded from the domain. Every other real number is allowed.

This note focuses on the function's end behavior — what happens to $f(x)$ as $x \to \pm \infty$ — which depends on comparing the degree of the numerator to the degree of the denominator.

The Three End Behavior Cases

CONCEPT

Case 1: Numerator Degree < Denominator Degree

The bottom grows faster, so the fraction shrinks toward zero.

\lim_{x \to \pm\infty} f(x) = 0

CONCEPT

Case 2: Numerator Degree = Denominator Degree

Both grow at the same rate — the fraction settles at the ratio of leading coefficients.

\lim_{x \to \pm\infty} f(x) = \frac{\text{leading coeff of num}}{\text{leading coeff of denom}}

CONCEPT

Case 3: Numerator Degree > Denominator Degree

There is NO horizontal asymptote — but that does NOT mean end behavior doesn't exist!

The function still goes somewhere as $x \to \pm \infty$ — it just doesn't settle onto a flat horizontal line. Instead, it grows without bound (like a polynomial) or follows another curve entirely.

The simplest case (numerator exactly one degree higher) follows a slant line — see 1.7B. When the gap is two or more degrees, the end behavior follows a curve (parabola, cubic, etc.) instead of a line, found the same way: by long division.

KEY RULE

"No horizontal asymptote" ≠ "no end behavior." The function still goes somewhere — it just isn't a flat line.

$f(x) = (2x+1)/(x-3)$ : equal degrees give a horizontal asymptote at the ratio of leading coefficients, $y = 2$ .

Quick check

What is the horizontal asymptote of $g(x) = \frac{4 - 9x^{2}}{3x^{2} + 2x}$ ?

Real-Life Examples

REAL-LIFE EXAMPLE

Where This Shows Up

Mixing a solution: if you keep adding pure water to a fixed amount of salt, the concentration approaches — but never quite reaches — zero. This is a horizontal asymptote at $y = 0$ .

Average cost per unit: if a factory has a fixed setup cost plus a cost per item, average cost per item settles toward the per-item cost as production scales up hugely — a horizontal asymptote based on equal degrees.

Step-by-Step: Numerator Degree Less Than Denominator

Worked example

Find the horizontal asymptote of $f(x) = (3x + 2) / (x^{2} + 1)$ , using limit notation.

f(x) = \frac{3x + 2}{x^2 + 1}

01
Compare degrees: numerator has degree 1, denominator has degree 2.
02
Since the numerator's degree is smaller, the horizontal asymptote is $y = 0$ :
$\lim_{x \to \pm\infty} f(x) = 0$
03
This makes sense: as $x$ grows huge, the $x^{2}$ in the denominator overwhelms the $x$ in the numerator, shrinking the whole fraction toward 0.

$f(x) = (3x+2)/(x^{2}+1)$ settles toward $y = 0$ in both directions.

Step-by-Step: Equal Degrees

Worked example

Find the horizontal asymptote of $f(x) = (2x + 1) / (x - 3)$ , using limit notation.

f(x) = \frac{2x + 1}{x - 3}

01
Compare degrees: both numerator and denominator have degree 1 — they are equal.
02
Take the ratio of the leading coefficients: 2 (from 2x) divided by 1 (from x):
$\lim_{x \to \pm\infty} f(x) = \frac{2}{1} = 2$
03
The horizontal asymptote is $y = 2$ .

COMMON MISTAKE

Don't confuse a horizontal asymptote with a value the function can never reach at any $x$ — a function can actually cross its horizontal asymptote for some x-values; the asymptote only describes far-away behavior.

Always fully compare degrees first — don't just look at the coefficients before checking which case you're in.

"No horizontal asymptote" does NOT mean "no end behavior." The function still has a definite end behavior in Case 3 — it just isn't a flat horizontal line.

Quick check

Can the graph of a rational function cross its horizontal asymptote?

Practice Problems

Worked example

Problem 1. Find the horizontal asymptote of $f(x) = (5x^{2}) / (2x^{2} + 3)$ .

f(x) = \frac{5x^2}{2x^2 + 3}

01
Degrees are equal (both degree 2).
02
Ratio of leading coefficients:
$\lim_{x \to \pm\infty} f(x) = \frac{5}{2}$

Worked example

Problem 2. Find the horizontal asymptote of $g(x) = 4 / (x + 1)$ .

g(x) = \frac{4}{x + 1}

01
Numerator degree (0, since 4 is a constant) is less than denominator degree (1).
02
Horizontal asymptote:
$\lim_{x \to \pm\infty} g(x) = 0$

Common slips

Don't confuse a horizontal asymptote with a value the function can never reach at any $x$ — a function can actually cross its horizontal asymptote for some x-values; the asymptote only describes far-away behavior.

Always fully compare degrees first — don't just look at the coefficients before checking which case you're in.

"No horizontal asymptote" does not mean "no end behavior." The function still has a definite end behavior in Case 3 — it just isn't a flat horizontal line.

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14 cards · Rational functions, Horizontal asymptote or end behavior?

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Recap card

5 lines to re-read the night before.

01
Compare the degree of the numerator to the degree of the denominator to find horizontal asymptote behavior.
02
Numerator < denominator → $y = 0$ .
03
Equal degrees → y = ratio of leading coefficients.
04
Numerator > denominator → no horizontal asymptote (see 1.7B).
05
Limit notation — lim( $x \to \pm \infty$ ) $f(x)$ — is the formal way to state a rational function's end behavior.

Practice 24 problems Next: Chapter 1.7B Rational Functions and End Behavior

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Rational Functions and End Behavior

What Is a Rational Function?

The Three End Behavior Cases

Real-Life Examples

Step-by-Step: Numerator Degree Less Than Denominator

Step-by-Step: Equal Degrees

Practice Problems

Recap card

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