AP®MathsCollege BoardPrecalculusRevision NotesPolynomial & Rational FunctionsRational FunctionsEnd Behavior of Rational Functions

End Behavior of Rational Functions (College Board AP® Precalculus): Revision Note

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 9 April 2026

End behavior of rational functions

How is the end behavior of a rational function determined?

The end behavior of a rational function depends on how the degrees of the numerator and denominator compare
- Remember that the degree of a polynomial is the power of its highest power of $x$
- For input values of large magnitude (large and negative or large and positive)
  - a polynomial is dominated by its leading term
- Therefore, the end behavior of a rational function can be understood by
  - examining the quotient of the leading terms of the numerator and denominator

What if the numerator has a higher degree than the denominator?

If the numerator has a higher degree than the denominator
- then the numerator dominates
In that case the quotient of the leading terms is a nonconstant polynomial
- and the rational function has the same end behavior as that polynomial
  - i.e. it increases or decreases without bound
- There is no horizontal asymptote
- Be sure you are familiar with the end behavior of polynomial functions
E.g. for $r (x) = \frac{8 x^{5} + x - 2}{2 x^{2} - 5 x + 1}$ , the quotient of leading terms is $\frac{8 x^{5}}{2 x^{2}} = 4 x^{3}$
- so the end behavior is like $4 x^{3}$
  - i.e. $r$ decreases without bound 'to the left' (i.e. as $x$ decreases without bound)
  - and increases without bound 'to the right' (i.e. as $x$ increases without bound)

There is a special case if the numerator's degree exceeds the denominator's by exactly 1
- In this case the graph has a slant (oblique) asymptote
- This is a line parallel to the line given by the quotient of the leading terms
- The rational function still increases or decreases without bound (in the same way that the line does)
  - But as $x$ increases or decreases without bound
  - the graph of the rational function gets closer and closer to the line of the slant asymptote
- You can use polynomial long division to find the exact equation of a slant asymptote

Graph showing a curve y = r(x) approaching a slant asymptote, on x and y axes. The asymptote is labelled with an arrow. — **Example of a slant asymptote for a rational function**

E.g. for $r (x) = \frac{2 x^{3} + x}{x^{2} - 1}$ , the quotient of leading terms is $\frac{2 x^{3}}{x^{2}} = 2 x$ ,
- so the end behavior is like $2 x$
  - i.e. $r$ increases without bound 'to the right' (i.e. as $x$ increases without bound)
  - and decreases without bound 'to the left' (i.e. as $x$ decreases without bound)
- $r$ also has a slant asymptote parallel to $y = 2 x$

What if the numerator and denominator have the same degree?

If the numerator and denominator have the same degree
- then neither dominates
The quotient of the leading terms is a constant
The graph has a horizontal asymptote at $y = \frac{a_{n}}{b_{n}}$
- where $a_{n}$ and $b_{n}$ are the leading coefficients of the numerator and denominator respectively
- The rational function converges to a finite value as $x$ increases or decreases without bound
  - and the graph of the rational function gets closer and closer to the line of the horizontal asymptote

Graph of rational function y=r(x) with horizontal asymptote, showing curve behaviour near the horizontal asymptote. — **Example of a horizontal asymptote for a rational function**

E.g. for $r (x) = \frac{3 x^{2} + 1}{5 x^{2} - 2}$ , the quotient of leading terms is $\frac{3 x^{2}}{5 x^{2}} = \frac{3}{5}$
- so the graph of $r$ has a horizontal asymptote at $y = \frac{3}{5}$

What if the denominator has a higher degree than the numerator?

If the denominator has a higher degree than the numerator
- then the denominator dominates
In that case the quotient of the leading terms has a nonconstant polynomial in the denominator
- As $x$ increases or decreases without bound, the rational function approaches zero
- The graph has a horizontal asymptote at $y = 0$
  - I.e. the graph approaches $y = 0$ (the $x$ -axis) as $x$ increases or decreases without bound
E.g. for $r (x) = \frac{x + 4}{x^{3} - 2 x}$ , the quotient of leading terms is $\frac{x}{x^{3}} = \frac{1}{x^{2}}$
- $\frac{1}{x^{2}}$ approaches zero as $x$ increases or decreases without bound
- So the graph of $r$ has a horizontal asymptote at $y = 0$

How do I express rational function end behavior in limit notation?

When there is a horizontal asymptote at $y = b$
- $\lim_{x \to \infty} r (x) = b$ and $\lim_{x \to - \infty} r (x) = b$
  - The output values get arbitrarily close to $b$ and stay arbitrarily close as input values increase or decrease without bound
- This includes the case when the denominator dominates and the horizontal asymptote is at $y = 0$
  - In that case $\lim_{x \to \infty} r (x) = 0$ and $\lim_{x \to - \infty} r (x) = 0$
When the numerator dominates (no horizontal asymptote)
- The end behavior is determined by the quotient of the leading terms
  - following the rules for polynomial end behavior, applied to that quotient
- The limit notation for the end behaviors will therefore be of one of the following four forms
  - $\lim_{x \to - \infty} r (x) = - \infty$ and $\lim_{x \to \infty} r (x) = \infty$
  - $\lim_{x \to - \infty} r (x) = \infty$ and $\lim_{x \to \infty} r (x) = - \infty$
  - $\lim_{x \to - \infty} r (x) = \infty$ and $\lim_{x \to \infty} r (x) = \infty$
  - $\lim_{x \to - \infty} r (x) = - \infty$ and $\lim_{x \to \infty} r (x) = - \infty$

Examiner Tips and Tricks

Be careful when finding the leading terms of the numerator and denominator.

Remember that the leading term is the term with the highest power of $x$
It is not necessarily the term that is written first

Worked Example

The function $g$ is given by $g (x) = \frac{x^{3} - 7 x - 43}{3 - x}$ .

Determine the end behavior of $g$ as $x$ increases without bound.

Answer:

In $g$ the polynomial in the numerator dominates the polynomial in the denominator

because the highest power of $x$ in the numerator (3) is greater than the highest power of $x$ in the denominator (1)

Consider the quotient of the leading terms of the numerator and denominator

Be careful: 'leading term' doesn't mean 'first term as written'
It means the term with the highest power of $x$

$\frac{x^{3}}{- x} = - \frac{x^{2} \cdot x}{x} = - x^{2}$

As $x$ increases without bound, $g$ will have the same end behavior as $- x^{2}$

and $- x^{2}$ decreases without bound as $x$ increases without bound

Write this in proper limit notation

$\lim_{x \to \infty} g (x) = - \infty$

Examiner Tips and Tricks

On a calculator section of the exam you could also use your graphing calculator to help you explore the end behavior of a rational function.

But make sure that you can determine the end behavior analytically as well!

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End Behavior of Rational Functions (College Board AP® Precalculus): Revision Note

End behavior of rational functions

How is the end behavior of a rational function determined?

What if the numerator has a higher degree than the denominator?

What if the numerator and denominator have the same degree?

What if the denominator has a higher degree than the numerator?

How do I express rational function end behavior in limit notation?

Examiner Tips and Tricks

Worked Example

Examiner Tips and Tricks

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Mark Curtis