In calculus, a one-sided limit refers to either one of the two

limits
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of a function $f(x)$ of a real variable $x$ as $x$ approaches a specified point either from the left or from the right.
The limit as $x$ decreases in value approaching $a$ ($x$ approaches $a$ "from the right" or "from above") can be denoted:
$$\backslash lim\_f(x)\; \backslash quad\; \backslash text\; \backslash quad\; \backslash lim\_\backslash ,f(x)\; \backslash quad\; \backslash text\; \backslash quad\; \backslash lim\_\backslash ,f(x)\; \backslash quad\; \backslash text\; \backslash quad\; f(x+)$$
The limit as $x$ increases in value approaching $a$ ($x$ approaches $a$ "from the left" or "from below") can be denoted:
$$\backslash lim\_f(x)\; \backslash quad\; \backslash text\; \backslash quad\; \backslash lim\_\backslash ,\; f(x)\; \backslash quad\; \backslash text\; \backslash quad\; \backslash lim\_\backslash ,f(x)\; \backslash quad\; \backslash text\; \backslash quad\; f(x-)$$
If the limit of $f(x)$ as $x$ approaches $a$ exists then the limits from the left and from the right both exist and are equal. In some cases in which the limit
$$\backslash lim\_\; f(x)$$
does not exist, the two one-sided limits nonetheless exist. Consequently, the limit as $x$ approaches $a$ is sometimes called a "two-sided limit".
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

Definition

If $I$ represents some interval that is contained in thedomain
Domain may refer to:
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of $f$ and if $a$ is point in $I$ then the right-sided limit as $x$ approaches $a$ can be rigorously defined as the value $R$ that satisfies:
$$\backslash text\; \backslash varepsilon\; >\; 0\backslash ;\backslash text\; \backslash delta\; >\; 0\; \backslash ;\backslash text\; x\; \backslash in\; I,\; \backslash text\; \backslash ;0\; <\; x\; -\; a\; <\; \backslash delta\; \backslash text\; ,\; f(x)\; -\; R,\; <\; \backslash varepsilon,$$
and the left-sided limit as $x$ approaches $a$ can be rigorously defined as the value $L$ that satisfies:
$$\backslash text\; \backslash varepsilon\; >\; 0\backslash ;\backslash text\; \backslash delta\; >\; 0\; \backslash ;\backslash text\; x\; \backslash in\; I,\; \backslash text\; \backslash ;0\; <\; a\; -\; x\; <\; \backslash delta\; \backslash text\; ,\; f(x)\; -\; L,\; <\; \backslash varepsilon.$$
We can represent the same thing more symbolically, as follows.
Let $I$ represent an interval, where $I\; \backslash subseteq\; \backslash mathrm(f)$, and $a\; \backslash in\; I$.
:$\backslash lim\_\; f(x)\; =\; R\; ~~~\; \backslash iff\; ~~~\; (\backslash forall\; \backslash varepsilon\; \backslash in\; \backslash mathbb\_,\; \backslash exists\; \backslash delta\; \backslash in\; \backslash mathbb\_,\; \backslash forall\; x\; \backslash in\; I,\; (0\; <\; x\; -\; a\; <\; \backslash delta\; \backslash longrightarrow\; ,\; f(x)\; -\; R\; ,\; <\; \backslash varepsilon))$
:$\backslash lim\_\; f(x)\; =\; L\; ~~~\; \backslash iff\; ~~~\; (\backslash forall\; \backslash varepsilon\; \backslash in\; \backslash mathbb\_,\; \backslash exists\; \backslash delta\; \backslash in\; \backslash mathbb\_,\; \backslash forall\; x\; \backslash in\; I,\; (0\; <\; a\; -\; x\; <\; \backslash delta\; \backslash longrightarrow\; ,\; f(x)\; -\; L\; ,\; <\; \backslash varepsilon))$
Intuition

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: :$\backslash lim\_\; f(x)\; =\; L\; ~~~\; \backslash iff\; ~~~\; \backslash forall\; \backslash varepsilon\; \backslash in\; \backslash mathbb\_,\; \backslash exists\; \backslash delta\; \backslash in\; \backslash mathbb\_,\; \backslash forall\; x\; \backslash in\; I,\; 0\; <\; ,\; x\; -\; a,\; <\; \backslash delta\; \backslash implies\; ,\; f(x)\; -\; L\; ,\; <\; \backslash varepsilon$ To define a one-sided limit, we must modify this inequality. Note that the absolute distance between $x$ and $a$ is $,\; x\; -\; a,\; =\; ,\; (-1)(-x\; +\; a),\; =\; ,\; (-1)(a\; -\; x),\; =\; ,\; (-1),\; ,\; a\; -\; x,\; =\; ,\; a\; -\; x,$. For the limit from the right, we want $x$ to be to the right of $a$, which means that $a\; <\; x$, so $x\; -\; a$ is positive. From above, $x\; -\; a$ is the distance between $x$ and $a$. We want to bound this distance by our value of $\backslash delta$, giving the inequality $x\; -\; a\; <\; \backslash delta$. Putting together the inequalities $0\; <\; x\; -\; a$ and $x\; -\; a\; <\; \backslash delta$ and using the transitivity property of inequalities, we have the compound inequality $0\; <\; x\; -\; a\; <\; \backslash delta$. Similarly, for the limit from the left, we want $x$ to be to the left of $a$, which means that $x\; <\; a$. In this case, it is $a\; -\; x$ that is positive and represents the distance between $x$ and $a$. Again, we want to bound this distance by our value of $\backslash delta$, leading to the compound inequality $0\; <\; a\; -\; x\; <\; \backslash delta$. Now, when our value of $x$ is in its desired interval, we expect that the value of $f(x)$ is also within its desired interval. The distance between $f(x)$ and $L$, the limiting value of the left sided limit, is $,\; f(x)\; -\; L,$. Similarly, the distance between $f(x)$ and $R$, the limiting value of the right sided limit, is $,\; f(x)\; -\; R,$. In both cases, we want to bound this distance by $\backslash varepsilon$, so we get the following: $,\; f(x)\; -\; L,\; <\; \backslash varepsilon$ for the left sided limit, and $,\; f(x)\; -\; R,\; <\; \backslash varepsilon$ for the right sided limit.Examples

''Example 1'': The limits from the left and from the right of $g(x)\; :=\; -\; \backslash frac$ as $x$ approaches $a\; :=\; 0$ are $$\backslash lim\_\; =\; +\; \backslash infty\; \backslash qquad\; \backslash text\; \backslash qquad\; \backslash lim\_\; =\; -\; \backslash infty$$ The reason why $\backslash lim\_\; =\; +\; \backslash infty$ is because $x$ is always negative (since $x\; \backslash to\; 0^-$ means that $x\; \backslash to\; 0$ with all values of $x$ satisfying $x\; <\; 0$), which implies that $-\; 1/x$ is always positive so that $\backslash lim\_$ divergesA limit that is equal to $\backslash infty$ is said to verge to $\backslash infty$ rather than verge to $\backslash infty.$ The same is true when a limit is equal to $-\; \backslash infty.$ to $+\; \backslash infty$ (and not to $-\; \backslash infty$) as $x$ approaches $0$ from the left. Similarly, $\backslash lim\_\; =\; -\; \backslash infty$ since all values of $x$ satisfy $x\; >\; 0$ (said differently, $x$ is always positive) as $x$ approaches $0$ from the right, which implies that $-\; 1/x$ is always negative so that $\backslash lim\_$ diverges to $-\; \backslash infty.$ ''Example 2'': One example of a function with different one-sided limits is $f(x)\; =\; \backslash frac,$ (cf. picture) where the limit from the left is $\backslash lim\_\; f(x)\; =\; 0$ and the limit from the right is $\backslash lim\_\; f(x)\; =\; 1.$ To calculate these limits, first show that $$\backslash lim\_\; 2^\; =\; \backslash infty\; \backslash qquad\; \backslash text\; \backslash qquad\; \backslash lim\_\; 2^\; =\; 0$$ (which is true because $\backslash lim\_\; =\; +\; \backslash infty\; \backslash text\; \backslash lim\_\; =\; -\; \backslash infty$) so that consequently, $$\backslash lim\_\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac\; =\; 1$$ whereas $\backslash lim\_\; \backslash frac\; =\; 0$ because the denominator diverges to infinity; that is, because $\backslash lim\_\; 1\; +\; 2^\; =\; \backslash infty.$ Since $\backslash lim\_\; f(x)\; \backslash neq\; \backslash lim\_\; f(x),$ the limit $\backslash lim\_\; f(x)$ does not exist.Relation to topological definition of limit

The one-sided limit to a point $p$ 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 $p.$ Alternatively, one may consider the domain with ahalf-open interval topology
In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set \mathbb of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and has a number of inte ...

.
Abel's theorem

A noteworthy theorem treating one-sided limits of certain power series at the boundaries of their intervals of convergence is Abel's theorem.Notes

References

See also

* Projectively extended real line * Semi-differentiability *Limit superior and limit inferior
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a ...

{{Calculus topics
Real analysis
Limits (mathematics)
Functions and mappings