One-sided limit

approaches a specified point either from the left or from the right.

decreases in value approaching

increases in value approaching

is sometimes called a "two-sided limit".

[citation needed] 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.

represents some interval that is contained in the domain of

can be rigorously defined as the value

that satisfies:[6][verification needed]

can be rigorously defined as the value

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

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: To define a one-sided limit, we must modify this inequality.

Note that the absolute distance between

We want to bound this distance by our value of

Similarly, for the limit from the left, we want

that is positive and represents the distance between

Again, we want to bound this distance by our value of

, leading to the compound inequality

is in its desired interval, we expect that the value of

In both cases, we want to bound this distance by

for the left sided limit, and

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

picture) where the limit from the left is

To calculate these limits, first show that

because the denominator diverges to infinity; that is, because

The one-sided limit to a point

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

[1][verification needed] Alternatively, one may consider the domain with a half-open interval topology.

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