one sided

In calculus, a one-sided limit refers to either one of the two limits of a function



f
(
x
)


{\displaystyle f(x)}
of a real variable



x


{\displaystyle x}
as



x


{\displaystyle x}
approaches a specified point either from the left or from the right.The limit as



x


{\displaystyle x}
decreases in value approaching



a


{\displaystyle a}
(



x


{\displaystyle x}
approaches



a


{\displaystyle a}
"from the right" or "from above") can be denoted:
The limit as



x


{\displaystyle x}
increases in value approaching



a


{\displaystyle a}
(



x


{\displaystyle x}
approaches



a


{\displaystyle a}
"from the left" or "from below") can be denoted:
If the limit of



f
(
x
)


{\displaystyle f(x)}
as



x


{\displaystyle x}
approaches



a


{\displaystyle a}
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



x


{\displaystyle x}
approaches



a


{\displaystyle a}
is sometimes called a "two-sided limit".It is possible for exactly one of the two one-sided limits to exists (while the other does not exist). It is also possible for neither of the two one-sided limits to exists.

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