One sided limit

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

$\lim_f(x) \quad \text \quad \lim_\,f(x) \quad \text \quad \lim_\,f(x) \quad \text \quad f(x+)$

The limit as $x$ increases in value approaching $a$ ($x$ approaches $a$ "from the left" or "from below") can be denoted:

$\lim_f(x) \quad \text \quad \lim_\, f(x) \quad \text \quad \lim_\,f(x) \quad \text \quad f(x-)$

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
$\lim_ f(x)$
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 the domain 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:
$\text \varepsilon > 0\;\text \delta > 0 \;\text x \in I, \text \;0 < x - a < \delta \text , f(x) - R, < \varepsilon,$
and the left-sided limit as $x$ approaches $a$ can be rigorously defined as the value $L$ that satisfies:
$\text \varepsilon > 0\;\text \delta > 0 \;\text x \in I, \text \;0 < a - x < \delta \text , f(x) - L, < \varepsilon.$

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

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

:$\lim_ f(x) = R ~~~ \iff ~~~ (\forall \varepsilon \in \mathbb_, \exists \delta \in \mathbb_, \forall x \in I, (0 < x - a < \delta \longrightarrow , f(x) - R , < \varepsilon))$

:$\lim_ f(x) = L ~~~ \iff ~~~ (\forall \varepsilon \in \mathbb_, \exists \delta \in \mathbb_, \forall x \in I, (0 < a - x < \delta \longrightarrow , f(x) - L , < \varepsilon))$

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:

:$\lim_ f(x) = L ~~~ \iff ~~~ \forall \varepsilon \in \mathbb_, \exists \delta \in \mathbb_, \forall x \in I, 0 < , x - a, < \delta \implies , f(x) - L , < \varepsilon$

To define a one-sided limit, we must modify this inequality. Note that the absolute distance between $x$ 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 $\delta$, giving the inequality $x - a < \delta$. Putting together the inequalities $0 < x - a$ and $x - a < \delta$ and using the transitivity property of inequalities, we have the compound inequality $0 < x - a < \delta$.

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 $\delta$, leading to the compound inequality $0 < a - x < \delta$.

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 $\varepsilon$, so we get the following: $, f(x) - L, < \varepsilon$ for the left sided limit, and $, f(x) - R, < \varepsilon$ for the right sided limit.

Examples

''Example 1'':
The limits from the left and from the right of $g(x) := - \frac$ as $x$ approaches $a := 0$ are
$\lim_ = + \infty \qquad \text \qquad \lim_ = - \infty$
The reason why $\lim_ = + \infty$ is because $x$ is always negative (since $x \to 0^-$ means that $x \to 0$ with all values of $x$ satisfying $x < 0$), which implies that $- 1/x$ is always positive so that $\lim_$ divergesA limit that is equal to $\infty$ is said to verge to $\infty$ rather than verge to $\infty.$ The same is true when a limit is equal to $- \infty.$ to $+ \infty$ (and not to $- \infty$) as $x$ approaches $0$ from the left.
Similarly, $\lim_ = - \infty$ 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 $\lim_$ diverges to $- \infty.$

''Example 2'':
One example of a function with different one-sided limits is $f(x) = \frac,$ (cf. picture) where the limit from the left is $\lim_ f(x) = 0$ and the limit from the right is $\lim_ f(x) = 1.$
To calculate these limits, first show that
$\lim_ 2^ = \infty \qquad \text \qquad \lim_ 2^ = 0$
(which is true because $\lim_ = + \infty \text \lim_ = - \infty$)
so that consequently,
$\lim_ \frac = \frac = \frac = 1$
whereas
$\lim_ \frac = 0$ because the denominator diverges to infinity; that is, because $\lim_ 1 + 2^ = \infty.$
Since $\lim_ f(x) \neq \lim_ f(x),$ the limit $\lim_ f(x)$ 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 a half-open interval topology.

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

{{Calculus topics

Real analysis
Limits (mathematics)
Functions and mappings