Asymptote Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Asymptote.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An asymptote is a line that a curve approaches arbitrarily closely as the input (or output) grows without bound, but typically never reaches.

The graph gets infinitely close but never touches—like chasing something forever.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: An asymptote is a straight line a graph approaches without bound as the input or output grows.

Common stuck point: The procedure for asymptote is the easy part; the trap is believing a curve can never cross a horizontal asymptote. Asking "Is there a line the curve approaches arbitrarily closely as the input or output grows without bound?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is there a line the curve approaches arbitrarily closely as the input or output grows without bound?

Worked Examples

Example 1

easy

Find the vertical and horizontal asymptotes of

f(x) = \dfrac{3x}{x - 2}

Answer

Vertical asymptote:

x = 2

; Horizontal asymptote:

y = 3

First step

Vertical asymptote: set the denominator equal to zero.

x - 2 = 0 \Rightarrow x = 2

. Since the numerator

3(2)=6 \neq 0

, there is a vertical asymptote at

x = 2

Full solution

2
Horizontal asymptote: compare degrees. Both numerator and denominator have degree $1$ . Divide leading coefficients: $\frac{3}{1} = 3$ . Thus the horizontal asymptote is $y = 3$ .
3
Verify by taking the limit: $\lim_{x\to\infty}\frac{3x}{x-2} = \lim_{x\to\infty}\frac{3}{1-2/x} = 3$ .

Vertical asymptotes arise where the denominator is zero (and numerator is not). Horizontal asymptotes describe end behavior; when degrees are equal, the asymptote equals the ratio of leading coefficients.

Example 2

hard

Find the oblique (slant) asymptote of

g(x) = \dfrac{x^2 + x - 1}{x - 1}

Example 3

medium

Find all asymptotes of

f(x) = \dfrac{2x^2 - 3x}{x^2 - x - 2}

Example 4

medium

Find any holes or vertical asymptotes of

f(x) = \dfrac{x^2 - 9}{x - 3}

Example 5

medium

Find the asymptotes of

f(x) = \dfrac{x + 5}{x^2 - 25}

Example 6

medium

Find the slant (oblique) asymptote of

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

Example 7

hard

Find the slant asymptote of

f(x) = \dfrac{3x^2 - x + 4}{x + 2}

Example 8

hard

Find the slant asymptote of

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

Example 9

challenge

Find all asymptotes of

f(x) = \dfrac{x^2 + 1}{x^2 - 4} + \dfrac{1}{x}

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Identify all asymptotes of

h(x) = \dfrac{5}{x + 3}

Example 2

medium

For

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

, determine the horizontal asymptote and verify by computing

f(100)

Example 3

easy

Find the vertical asymptote of

f(x)=\dfrac{1}{x}

Example 4

easy

Find the horizontal asymptote of

f(x)=\dfrac{1}{x}

x\to\pm\infty

Example 5

easy

Does the polynomial

f(x)=x^2

have any asymptotes?

Example 6

easy

Find the vertical asymptote of

f(x)=\dfrac{1}{x-3}

Example 7

easy

The function

e^x

has which horizontal asymptote as

x\to-\infty

Example 8

easy

Can a function cross its horizontal asymptote?

Example 9

easy

At a vertical asymptote, what happens to the function's value?

Example 10

easy

Find the zero of

f(x)=\dfrac{x-2}{x+5}

Example 11

medium

Find the horizontal asymptote of

f(x)=\dfrac{3x}{x+1}

Example 12

medium

Find the horizontal asymptote of

f(x)=\dfrac{2x+1}{x^2-4}

Example 13

medium

Find the vertical asymptotes of

f(x)=\dfrac{x}{x^2-9}

Example 14

medium

Find the horizontal asymptote of

f(x)=\dfrac{4x^2-1}{2x^2+x}

Example 15

medium

Identify the removable discontinuity (hole) of

f(x)=\dfrac{x^2-1}{x-1}

Example 16

medium

Find the vertical and horizontal asymptotes of

f(x)=\dfrac{x+2}{x-4}

Example 17

medium

x\to\infty

, what value does

f(x)=2+\dfrac{5}{x}

approach?

Example 18

medium

Why does

\tan x

have vertical asymptotes at

x=\tfrac{\pi}{2}+k\pi

Example 19

medium

Find the horizontal asymptote of

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

Example 20

challenge

Find the slant (oblique) asymptote of

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

Example 21

challenge

For what value of

a

does

f(x)=\dfrac{x-a}{x^2-x-6}

have a hole rather than a vertical asymptote at

x=3

Example 22

challenge

A rational function crosses its horizontal asymptote

y=1

. Give such a function and the crossing point.

Example 23

easy

Find the horizontal asymptote of

f(x) = \dfrac{5x}{x + 2}

Example 24

easy

Does

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

have any vertical asymptotes?

Example 25

easy

What kind of asymptote does

f(x) = \ln(x)

have at

x = 0

Example 26

easy

Find all vertical asymptotes of

f(x) = \dfrac{1}{x^2 - 4}

Example 27

medium

Find the horizontal asymptote of

f(x) = \dfrac{7x^2 - 1}{2x^2 + 3x}

Example 28

medium

Find the horizontal asymptote of

f(x) = \dfrac{4x^3 + 1}{2x^3 - 3x}

x \to \pm\infty

Example 29

medium

Does

f(x) = \dfrac{3x^2 + 1}{x + 5}

have a horizontal asymptote?

Example 30

medium

Find the vertical asymptotes of

f(x) = \dfrac{x}{x^2 - x - 6}

Example 31

medium

What is the horizontal asymptote of

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

x \to \infty

Example 32

hard

For what value of

a

does

f(x) = \dfrac{ax^2 + 1}{x^2 - 4}

have horizontal asymptote

y = 3

Example 33

hard

Find the asymptotes of

f(x) = \tan x

\left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right)

Example 34

hard

Find the value of

b

so that

f(x) = \dfrac{x^2 + b}{x - 1}

has a hole at

x = 1

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Related Concepts

Function Limit Infinity

Background Knowledge

These ideas may be useful before you work through the harder examples.

function definition