Limit of a function

In calculus, a branch of mathematics, the limit of a function is the behavior of a certain function near a selected input value for that function. Limits are one of the main calculus topics, along with derivatives, integration, and differential equations.

Definition of the limit

The definition of the limit is as follows:

If the function [math]\displaystyle{ f(x) }[/math] approaches a number [math]\displaystyle{ L }[/math] as [math]\displaystyle{ x }[/math] approaches a number [math]\displaystyle{ c }[/math], then [math]\displaystyle{ \lim_{x \to c}f(x) = L. }[/math]

The notation for the limit above is read as "The limit of [math]\displaystyle{ f(x) }[/math] as [math]\displaystyle{ x }[/math] approaches [math]\displaystyle{ c }[/math] is [math]\displaystyle{ L }[/math]", or alternatively, [math]\displaystyle{ f(x) \to L }[/math] as [math]\displaystyle{ x \to c }[/math] (reads "[math]\displaystyle{ f(x) }[/math] tends to [math]\displaystyle{ L }[/math] as [math]\displaystyle{ x }[/math] tends to [math]\displaystyle{ c }[/math]"^[1]). Informally, this means that we can make [math]\displaystyle{ f(x) }[/math] as close to [math]\displaystyle{ L }[/math] as possible—by making [math]\displaystyle{ x }[/math] sufficiently close to [math]\displaystyle{ c }[/math] from both sides (without making [math]\displaystyle{ x }[/math] equal to [math]\displaystyle{ c }[/math]).^[2]

Imagine we have a function such as [math]\displaystyle{ f(x)=1/x }[/math]. When [math]\displaystyle{ x=0 }[/math], [math]\displaystyle{ f(x) }[/math] is undefined, because [math]\displaystyle{ f(0)=1/0 }[/math]. Therefore, on the Cartesian coordinate system, the function [math]\displaystyle{ f(x)=1/x }[/math] would have a vertical asymptote at [math]\displaystyle{ x=0 }[/math]. In limit notation, this would be written as:

The limit of [math]\displaystyle{ 1/x }[/math] as [math]\displaystyle{ x }[/math] approaches [math]\displaystyle{ 0 }[/math] is [math]\displaystyle{ \infty }[/math], which is denoted by [math]\displaystyle{ \lim_{x \to 0}1/x = \infty. }[/math]

Right and left limits

For the function [math]\displaystyle{ f(x)=1/x }[/math], we can get as close to [math]\displaystyle{ 0 }[/math] in the [math]\displaystyle{ x }[/math]-values as we want, so long as we do not make [math]\displaystyle{ x }[/math] equal to [math]\displaystyle{ 0 }[/math]. For instance, we could make x=.00000001 or -.00000001, but never 0. Therefore, we can get [math]\displaystyle{ f(x) }[/math] as close as we want to [math]\displaystyle{ \infty }[/math], but without reaching it.^[3] The left limit is any value that approaches the limit from numbers less than the number, and the right limit is any value that approaches the limit from number greater than the limit number. For instance, in the function [math]\displaystyle{ f(x)=1/x }[/math], since the limit for [math]\displaystyle{ x }[/math] is 0, if [math]\displaystyle{ x=1 }[/math], it approaches the limit from the right. If we instead choose -1, we say it approaches the limit from the left.

Limit Of A Function Media

For the depicted $f$ , $a$ , and $b$ , we can ensure that the value $f (x)$ is within an arbitrarily small interval $(b - ε, b + ε)$ by restricting $x$ to a sufficiently small interval $(a - δ, a + δ).$ Hence $f (x) \to b$ as $x \to a$ .
The limit as x \to x_0^+ differs from that as x \to x_0^-. Therefore, the limit as $x \to x 0$ does not exist.
Function without a limit at an essential discontinuity
The limit of this function at infinity exists
Horizontal asymptote about $y = 4$