One-sided limit explained

f(x) of a real variable x as x approaches a specified point either from the left or from the right.^{[1] [2]}

The limit as

x decreases in value approaching a (x approaches a "from the right"^[3] 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"^[4] 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 the domain of f and if a is a point in I then the right-sided limit as x approaches a can be rigorously defined as the value R that satisfies:^[5] and the left-sided limit as x approaches a can be rigorously defined as the value L that satisfies:

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

Let

I represent an interval, where I \subseteq domain (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) : = - (1)/(x) as x approaches a : = 0 are$$\backslash lim\_\; =\; +\; \backslash infty\; \backslash qquad\; \backslash text\; \backslash qquad\; \backslash lim\_\; =\; -\; \backslash infty$$The reason why \lim _{x \to 0 ^{-}} { -1 /x} = + inf ty 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 _{x \to 0 ^{-}} { -1 /x} diverges^[6] to + inf ty (and not to - inf ty ) as x approaches 0 from the left. Similarly, \lim _{x \to 0 ^{+}} { -1 /x} = - inf ty 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 _{x \to 0 ^{+}} { -1 /x} diverges to - inf ty.

Example 2: One example of a function with different one-sided limits is

f(x) = (1)/(1+2 ^{-1/x}) , (cf. picture) where the limit from the left is \lim _{x \to 0 ^{-}} f(x) = 0 and the limit from the right is \lim _{x \to 0 ^{+}} 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 \lim _{x \to 0 ^{-}} { -1 /x} = + inf ty and \lim _{x \to 0 ^{+}} { -1 /x} = - inf ty )so that consequently, $$\backslash lim\_\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac\; =\; 1$$whereas \lim _{x \to 0 ^{-}} (1)/(1+2 ^{-1/x}) = 0 because the denominator diverges to infinity; that is, because \lim _{x \to 0 ^{-}} 1 + 2 ^{-1/x} = inf ty. Since \lim _{x \to 0 ^{-}} f(x) ≠ \lim _{x \to 0 ^{+}} f(x), the limit \lim _{x} f(x) does not exist.

Relation to topological definition of limit

See also: Filters in topology.

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

See main article: 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.

See also

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

Notes and References

1. Web site: One-sided limit - Encyclopedia of Mathematics. 7 August 2021. encyclopediaofmath.org.
2. Book: Fridy, J. A.. Introductory Analysis: The Theory of Calculus. 24 January 2020. Gulf Professional Publishing. 978-0-12-267655-0. 48. en. 7 August 2021.
3. Hasan. Osman. Khayam. Syed. 2014-01-02. Towards Formal Linear Cryptanalysis using HOL4. Journal of Universal Computer Science. en. 20. 2. 209. 10.3217/jucs-020-02-0193. 0948-6968.
4. Phase Phenomena of Proteins in Living Matter. 2020-12-12. Thesis. en. Andrei G.. Gasic.
5. Book: Giv, Hossein Hosseini. Mathematical Analysis and Its Inherent Nature. 28 September 2016. American Mathematical Soc.. 978-1-4704-2807-5. 130. en. 7 August 2021.
6. A limit that is equal to inf ty is said to verge to inf ty rather than verge to inf ty. The same is true when a limit is equal to - inf ty.