AP®MathsCollege BoardCalculus BCStudy GuidesUnit 1: Limits & ContinuityLimitsAsymptotes

Asymptotes (College Board AP® Calculus BC): Study Guide

Written by: Roger B

Reviewed by: Dan Finlay

Updated on 1 May 2026

Horizontal asymptote

What is a horizontal asymptote?

A horizontal asymptote is a horizontal line
- that the graph of a function gets closer and closer to (but never touches or intersects)
- as x becomes unbounded in the positive or negative direction
On the following diagram, the horizontal asymptote is indicated by a dashed line

A graph of a function f with a horizontal asymptote at y=c — **An example of a function with a horizontal asymptote at y=c**

How can I identify horizontal asymptotes using limits?

A function will have a horizontal asymptote if it has a finite limit at infinity
- I.e. the line $y = c$ will be a horizontal asymptote for the graph of a function $f$ if
  - $\lim_{x \to \infty} f (x) = c$ , or
  - $\lim_{x \to - \infty} f (x) = c$
Horizontal asymptotes (if any) may therefore be determined by evaluating the limits at infinity

Examiner Tips and Tricks

By graphing a function on your graphing calculator you can:

spot any asymptotic behavior by a function at plus or minus infinity
check limits that you have determined analytically

How can I identify horizontal asymptotes numerically?

If a function has a horizontal asymptote, then its values will tend to a limit for large positive or negative values of $x$
Consider, the function $f (x) = \frac{12 + 5 e^{- \frac{x}{1000}}}{1 + e^{- \frac{x}{1000}}}$

Consider the values for large positive $x$

$x =$	$\frac{12 + 5 e^{- \frac{x}{1000}}}{1 + e^{- \frac{x}{1000}}} =$
100	8.674...
1,000	10.117...
10,000	11.999...
100,000	12

The table suggests that $\lim_{x \to \infty} f (x) = 12$
- Note $f (100 000) \neq 12$
- It starts as 11.999... with a large number of 9s
- Most calculators cannot differentiate between this value and 12
Consider the values for large negative $x$

$x =$	$\frac{12 + 5 e^{- \frac{x}{100}}}{1 + e^{- \frac{x}{100}}} =$
-100	8.325...
-1,000	6.882...
-10,000	5.000...
-100,000	5

The table suggests that $\lim_{x \to - \infty} f (x) = 5$
- Note $f (- 100 000) \neq 5$
- It starts as 5.000... with a large number of 0s
- Most calculators cannot differentiate between this value and 5

Vertical asymptote

What is a vertical asymptote?

A vertical asymptote is a vertical line
- that the graph of a function gets closer and closer to (but never touches or intersects)
- as x gets closer and closer to the x-value of the vertical line
On the following diagram, the vertical asymptote is indicated by a dashed line

Graph of a function f with a vertical asymptote at x=c — **An example of a function with a vertical asymptote at x=c**

How can I identify vertical asymptotes using limits?

A function will have a vertical asymptote at any x-value where the function becomes unbounded
- I.e. the line $x = c$ will be a vertical asymptote for the graph of a function $f$ if
  - $\lim_{x \to c^{-}} f (x) = \pm \infty$ , or
  - $\lim_{x \to c^{+}} f (x) = \pm \infty$
Vertical asymptotes (if any) may therefore be determined by identifying points where the function becomes unbounded
- Usually this will involve a function in the form of a quotient
  - at points where the denominator becomes zero

How can I identify vertical asymptotes numerically?

If a function has a vertical asymptote, then its values tend to positive or negative infinity as the values of $x$ get closer to the vertical asymptote
Consider, the function $f (x) = \csc x$
Consider the values for values of $x$ just smaller than 0

$x =$	$\csc x =$
-0.1	-10.016...
-0.01	-100.001...
-0.001	-1,000.000...
-0.0001	-10,000.000...

The table suggests that $\lim_{x \to 0^{-}} f (x) = - \infty$
- The values of $f$ keep getting further from 0 as $x$ gets closer to 0
Consider the values for values of $x$ just bigger than 0

$x =$	$\csc x =$
0.1	10.016...
0.01	100.001...
0.001	1,000.000...
0.0001	10,000.000...

The table suggests that $\lim_{x \to 0^{+}} f (x) = \infty$
- The values of $f$ keep getting further from 0 as $x$ gets closer to 0

Worked Example

Let $f$ be the function defined by $f (x) = \frac{3 x - 11}{x - 2}$ .

Using limits, identify the vertical and horizontal asymptotes (if any) on the graph of $f$ .

Answer:

The denominator becomes 0 when $x = 2$ , so start by considering the limits there

At 2 the numerator is equal to -5, so zero only occurs in the denominator

Just 'to the left' of 2, $3 x - 11 < 0$ and $x - 2 < 0$ so

$\lim_{x \to 2^{-}} f (x) = \infty$

Just 'to the right' of 2, $3 x - 11 < 0$ and $x - 2 > 0$ so

$\lim_{x \to 2^{+}} f (x) = - \infty$

This confirms that the graph of $f$ has a vertical asymptote at $x = 2$

To identify horizontal asymptotes, start by rearranging to make the behavior of the function more obvious

$\frac{3 x - 11}{x - 2} = \frac{3 (x - 2) - 5}{x - 2} = \frac{3 (x - 2)}{x - 2} - \frac{5}{x - 2} = 3 - \frac{5}{x - 2}$

$\frac{5}{x - 2}$ becomes closer and closer to zero as x increases in the positive or negative directions, so

$\lim_{x \to - \infty} f (x) = \lim_{x \to \infty} f (x) = 3 - 0 = 3$

Alternatively, divide the top and bottom of the function by $x$

$\lim_{x \to \infty} f (x) = \lim_{x \to \infty} \frac{3 - \frac{11}{x}}{1 - \frac{2}{x}} = \frac{3}{1} = 3$

$\lim_{x \to - \infty} f (x) = \lim_{x \to - \infty} \frac{3 - \frac{11}{x}}{1 - \frac{2}{x}} = \frac{3}{1} = 3$

This means that the graph of $f$ has a horizontal asymptote at $y = 3$

The graph of $f$ has a vertical asymptote at $x = 2$ , and a horizontal asymptote at $y = 3$

Unlock more, it's free!

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

the (exam) results speak for themselves:

I would just like to say a massive thank you for putting together such a brilliant, easy to use website.I really think using this site helped me secure my top gradesin science and maths. You really did save my exams! Thank you.

Beth
IGCSE Student

This website is soooo useful and I can’t ever thank you enough for organising questions by topic like this. Furthermore, the name of the website could not have been more appropriate as it literally did SAVE MY EXAMS!

Fathima
A Level Student

Incredible! SO worth my money, the revision notes have everything I need to know and are so easy to understand. I actually enjoy revising! It makes me feel a lot more confident for my GCSEs in a few months.

Kate
GCSE Student

Absolutely brilliant, both my girls used it for A levels and GCSE. It's saves on paper copies, also beneficial exam questions ranked from easy to hard. It's removed a lot of stress from the exams.

Sameera
Parent

Just to say that your resources are the best I have seen and I have been teaching chemistry at different levels for about 40 years

Mark
Chemistry Teacher

Excellent