One-sided limit

One-sided limit
In calculus, a one-sided limit refers to either one of the two limits of a function of a real variable as approaches a specified point either from the left or from the right.
The limit as decreases in value approaching (approaches "from the right" or "from above") can be denoted:
The limit as increases in value approaching (approaches "from the left" or "from below") can be denoted:
If the limit of as approaches exists then the limits from the left and from the right both exist and are equal. In some cases in which the limit
does not exist, the two one-sided limits nonetheless exist. Consequently, the limit as approaches 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 represents some interval that is contained in the domain of and if is a point in then the right-sided limit as approaches can be rigorously defined as the value that satisfies:
and the left-sided limit as approaches can be rigorously defined as the value that satisfies:
We can represent the same thing more symbolically, as follows.
Let represent an interval, where , and .
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:
To define a one-sided limit, we must modify this inequality. Note that the absolute distance between and is .
For the limit from the right, we want to be to the right of , which means that , so is positive. From above, is the distance between and . We want to bound this distance by our value of , giving the inequality . Putting together the inequalities and and using the transitivity property of inequalities, we have the compound inequality .
Similarly, for the limit from the left, we want to be to the left of , which means that . In this case, it is that is positive and represents the distance between and . Again, we want to bound this distance by our value of , leading to the compound inequality .
Now, when our value of is in its desired interval, we expect that the value of is also within its desired interval. The distance between and , the limiting value of the left sided limit, is . Similarly, the distance between and , the limiting value of the right sided limit, is . In both cases, we want to bound this distance by , so we get the following: for the left sided limit, and for the right sided limit.
Examples
Example 1:
The limits from the left and from the right of as approaches are
The reason why is because is always negative (since means that with all values of satisfying ), which implies that is always positive so that diverges to (and not to ) as approaches from the left.
Similarly, since all values of satisfy (said differently, is always positive) as approaches from the right, which implies that is always negative so that diverges to
Example 2:
One example of a function with different one-sided limits is (cf. picture) where the limit from the left is and the limit from the right is
To calculate these limits, first show that
(which is true because )
so that consequently,
whereas
because the denominator diverges to infinity; that is, because
Since the limit does not exist.
Relation to topological definition of limit
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 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
Real analysis
Limits (mathematics)
Functions and mappings