In calculus, a one-sided limit refers to either one of the two limits of a function of a real approaches a specified point either from the left or from the right.

"from the right" or "from above"), is denoted:

"from the left" or "from below"), is denoted:

is sometimes called a "two-sided limit".

In some cases in which the two-sided limit does not exist, the two individual one-sided limits nonetheless exist and they are then necessarily unequal.

It is possible for only one of the two one-sided limits to exist. It is also possible for neither of the two one-sided limits to exist.

Formal definition

Definition

represents some interval that is contained in the domain of a function that satisfies: that satisfies:

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:

is

and using the transitivity.

.

for the right sided limit.

Examples

Example 1. diverges

Example 2. does not exist.

Relation to topological definition of limit

corresponds to the general definition of limit 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