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Concept# One-sided limit

Summary

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.
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 .
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 .

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