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One-sided limit

Limit of a function approaching a value point from values below or above the value point

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Incalculus, a one-sided limit refers to either one of the two limits of a function ${\displaystyle f($x)} of a real variable ${\displaystyle x}$as${\displaystyle x}$ approaches a specified point either from the left or from the right.^[1][2]

The limit as ${\displaystyle x}$ decreases in value approaching ${\displaystyle a}$ (${\displaystyle x}$ approaches ${\displaystyle a}$ "from the right"^[3] or "from above") can be denoted:^[1][2]

The limit as ${\displaystyle x}$ increases in value approaching ${\displaystyle a}$ (${\displaystyle x}$ approaches ${\displaystyle a}$ "from the left"^[4][5] or "from below") can be denoted:^[1][2]

If the limit of ${\displaystyle f($x)} as${\displaystyle x}$ approaches ${\displaystyle a}$ exists then the limits from the left and from the right both exist and are equal. In some cases in which the limit

${\displaystyle x}$

${\displaystyle a}$

It is possible for exactly one of the two one-sided limits to exist (while the other does not exist). It is also possible for neither of the two one-sided limits to exist.

Formal definition[edit]

Definition[edit]

If${\displaystyle I}$ represents some interval that is contained in the domainof${\displaystyle f}$ and if ${\displaystyle a}$ is a point in ${\displaystyle I}$ then the right-sided limit as ${\displaystyle x}$ approaches ${\displaystyle a}$ can be rigorously defined as the value ${\displaystyle R}$ that satisfies:^{[6][verification needed]}

${\displaystyle x}$

${\displaystyle a}$

${\displaystyle L}$

We can represent the same thing more symbolically, as follows.

Let ${\displaystyle I}$ represent an interval, where ${\displaystyle I\subseteq \mathrm {domain} ($f)} , and ${\displaystyle a\in I}$.

$${\displaystyle \lim _{x\to a^{+}}f($$x)=R~~~\iff ~~~(\forall \varepsilon \in \mathbb {R} _{+},\exists \delta \in \mathbb {R} _{+},\forall x\in I,(0<x-a<\delta \longrightarrow |f(x)-R|<\varepsilon ))}

$${\displaystyle \lim _{x\to a^{-}}f($$x)=L~~~\iff ~~~(\forall \varepsilon \in \mathbb {R} _{+},\exists \delta \in \mathbb {R} _{+},\forall x\in I,(0<a-x<\delta \longrightarrow |f(x)-L|<\varepsilon ))}

Intuition[edit]

In comparison to the formal definition for the limit of a function at a point, the one-sided limit (as the name would suggest) only deals with input values to one side of the approached input value.

For reference, the formal definition for the limit of a function at a point is as follows:

$${\displaystyle \lim _{x\to a}f($$x)=L~~~\iff ~~~\forall \varepsilon \in \mathbb {R} _{+},\exists \delta \in \mathbb {R} _{+},\forall x\in I,0<|x-a|<\delta \implies |f(x)-L|<\varepsilon .}

To define a one-sided limit, we must modify this inequality. Note that the absolute distance between ${\displaystyle x}$ and ${\displaystyle a}$is

For the limit from the right, we want ${\displaystyle x}$ to be to the right of ${\displaystyle a}$, which means that ${\displaystyle a<x}$, so ${\displaystyle x-a}$ is positive. From above, ${\displaystyle x-a}$ is the distance between ${\displaystyle x}$ and ${\displaystyle a}$. We want to bound this distance by our value of ${\displaystyle \delta }$, giving the inequality ${\displaystyle x-a<\delta }$. Putting together the inequalities ${\displaystyle 0<x-a}$ and ${\displaystyle x-a<\delta }$ and using the transitivity property of inequalities, we have the compound inequality ${\displaystyle 0<x-a<\delta }$.

Similarly, for the limit from the left, we want ${\displaystyle x}$ to be to the left of ${\displaystyle a}$, which means that ${\displaystyle x<a}$. In this case, it is ${\displaystyle a-x}$ that is positive and represents the distance between ${\displaystyle x}$ and ${\displaystyle a}$. Again, we want to bound this distance by our value of ${\displaystyle \delta }$, leading to the compound inequality ${\displaystyle 0<a-x<\delta }$.

Now, when our value of ${\displaystyle x}$ is in its desired interval, we expect that the value of ${\displaystyle f($x)} is also within its desired interval. The distance between ${\displaystyle f($x)} and ${\displaystyle L}$, the limiting value of the left sided limit, is ${\displaystyle |f($x)-L|} . Similarly, the distance between ${\displaystyle f($x)} and ${\displaystyle R}$, the limiting value of the right sided limit, is ${\displaystyle |f($x)-R|} . In both cases, we want to bound this distance by ${\displaystyle \varepsilon }$, so we get the following: ${\displaystyle |f($x)-L|<\varepsilon } for the left sided limit, and ${\displaystyle |f($x)-R|<\varepsilon } for the right sided limit.

Examples[edit]

Example 1: The limits from the left and from the right of ${\displaystyle g($x):=-{\frac {1}{x}}} as${\displaystyle x}$ approaches ${\displaystyle a:=0}$ are

${\displaystyle \lim _{x\to 0^{-}}{-1/x}=+\infty }$

${\displaystyle x}$

${\displaystyle x\to 0^{-}}$

${\displaystyle x\to 0}$

${\displaystyle x}$

${\displaystyle x<0}$

${\displaystyle -1/x}$

${\displaystyle \lim _{x\to 0^{-}}{-1/x}}$

${\displaystyle +\infty }$

${\displaystyle -\infty }$

${\displaystyle x}$

${\displaystyle 0}$

${\displaystyle \lim _{x\to 0^{+}}{-1/x}=-\infty }$

${\displaystyle x}$

${\displaystyle x>0}$

${\displaystyle x}$

${\displaystyle x}$

${\displaystyle 0}$

${\displaystyle -1/x}$

${\displaystyle \lim _{x\to 0^{+}}{-1/x}}$

${\displaystyle -\infty .}$

Example 2: One example of a function with different one-sided limits is ${\displaystyle f($x)={\frac {1}{1+2^{-1/x}}},} (cf. picture) where the limit from the left is ${\displaystyle \lim _{x\to 0^{-}}f($x)=0} and the limit from the right is ${\displaystyle \lim _{x\to 0^{+}}f($x)=1.} To calculate these limits, first show that

${\displaystyle \lim _{x\to 0^{-}}{-1/x}=+\infty {\text{ and }}\lim _{x\to 0^{+}}{-1/x}=-\infty }$

${\displaystyle \lim _{x\to 0^{-}}{\frac {1}{1+2^{-1/x}}}=0}$

${\displaystyle \lim _{x\to 0^{-}}1+2^{-1/x}=\infty .}$

${\displaystyle \lim _{x\to 0^{-}}f($x)\neq \lim _{x\to 0^{+}}f(x),}

${\displaystyle \lim _{x\to 0}f($x)}

Relation to topological definition of limit[edit]

The one-sided limit to a point ${\displaystyle p}$ corresponds to the general definition of limit, with the domain of the function restricted to one side, by either allowing that the function domain is a subset of the topological space, or by considering a one-sided subspace, including ${\displaystyle p.}$^{[1][verification needed]} Alternatively, one may consider the domain with a half-open interval topology.^{[citation needed]}

Abel's theorem[edit]

A noteworthy theorem treating one-sided limits of certain power series at the boundaries of their intervals of convergenceisAbel's theorem.^{[citation needed]}

Notes[edit]

References[edit]

See also[edit]

• Projectively extended real line
• Semi-differentiability
• Limit superior and limit inferior