asymptotes Definition
- 1a straight line that a curve approaches but never touches as it goes to infinity
- 2a line that limits a curve or function in such a way that as the independent variable approaches a certain value, the function approaches infinity or negative infinity or some finite limit
Using asymptotes: Examples
Take a moment to familiarize yourself with how "asymptotes" can be used in various situations through the following examples!
Example
The curve y = 1/x has two asymptotes, the x-axis and the y-axis.
Example
The function f(x) = 1/x has a vertical asymptote at x = 0.
Example
The hyperbola has two asymptotes that intersect at the center of the curve.
Phrases with asymptotes
a horizontal line that a curve approaches as the independent variable goes to infinity or negative infinity
Example
The function f(x) = 1/x has a horizontal asymptote at y = 0.
a vertical line that a curve approaches as the independent variable approaches a certain value
Example
The function f(x) = 1/x has a vertical asymptote at x = 0.
a slanted line that a curve approaches as the independent variable goes to infinity or negative infinity
Example
The function f(x) = x + 1 / x has an oblique asymptote at y = x.
Origins of asymptotes
from Greek 'asumptotos', meaning 'not falling together'
Summary: asymptotes in Brief
'Asymptotes' [ˈæsɪmptoʊts] are straight lines that a curve approaches but never touches as it goes to infinity. They limit a curve or function in such a way that as the independent variable approaches a certain value, the function approaches infinity or negative infinity or some finite limit. Phrases like 'horizontal asymptote,' 'vertical asymptote,' and 'oblique asymptote' describe different types of asymptotes.
