Practice Asymptote in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

medium

Find the vertical and horizontal asymptotes of

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

Example 2

easy

Identify all asymptotes of

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

Example 3

medium

Find the vertical asymptotes of

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

Example 4

medium

Find any holes or vertical asymptotes of

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

Example 5

medium

Does

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

have a horizontal asymptote?

Example 6

hard

Find the value of

b

so that

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

has a hole at

x = 1

Example 7

hard

For what value of

a

does

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

have horizontal asymptote

y = 3

Example 8

easy

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

Example 9

easy

Find the horizontal asymptote of

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

x\to\pm\infty

Example 10

easy

Does

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

have any vertical asymptotes?

Example 11

challenge

Find all asymptotes of

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

Example 12

easy

Find all vertical asymptotes of

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

Example 13

easy

The function

e^x

has which horizontal asymptote as

x\to-\infty

Example 14

hard

Find the oblique (slant) asymptote of

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

Example 15

hard

Find both horizontal asymptotes of

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

Example 16

medium

Find all asymptotes of

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

Example 17

medium

Find the horizontal asymptote of

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

Example 18

easy

Find the zero of

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

Example 19

medium

Find the horizontal asymptote of

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

Example 20

medium

Find the horizontal asymptote of

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

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Related Concepts

Function Limit Infinity