One-sided limit

In calculus

Calculus

Calculus is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental...

, a one-sided limit is either of the two limits

Limit of a function

In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Informally, a function assigns an output f to every input x. The function has a limit L at an input p if f is "close" to L whenever x is...

 of a function

Function (mathematics)

In mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...

 f(x) of a real

Real number

In mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...

 variable x as x approaches a specified point either from below or from above. One should write either:
for the limit as x decreases in value approaching a (x approaches a "from the right" or "from above"), and similarly
for the limit as x increases in value approaching a (x approaches a "from the left" or "from below").

The two one-sided limits exist and are equal if and only if the limit of f(x) as x approaches a exists. In some cases in which the limit
does not exist, the two one-sided limits nonetheless exist. Consequently the limit as x approaches a is sometimes called a "two-sided limit". In some cases one of the two one-sided limits exists and the other does not, and in some cases neither exists.

Examples

One example of a function with different one-sided limits is the following:
whereas

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.

Abel's theorem

A noteworthy theorem treating one-sided limits of certain power series at the boundaries of their intervals of convergence

Radius of convergence

In mathematics, the radius of convergence of a power series is a non-negative quantity, either a real number or ∞, that represents a domain in which the series will converge. Within the radius of convergence, a power series converges absolutely and uniformly on compacta as well...

 is Abel's theorem

Abel's theorem

In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel.-Theorem:...

.