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One-sided limit

A one-sided limit refers to either of the two possible limits of a function $f x$ as $x$ approaches a certain point either from the left or from the right.[cite:@wikipedia2025onesided]

Notation

A one-sided limit "from the left" (as $x$ increases) is written

$\lim_{x \to a^{-}} f x,$

and a one-sided limit "from the right" (as $x$ decreases) is written

$\lim_{x \to a^{+}} f x .$

Existence

It is possible for neither one-sided limit to exist, for only one of the two limits to exist, or for both one-sided limits to exist (in which case, a proper limit does exist).

Evaluation

Extensionality

Similarly to two-sided limits, one-sided limits support a form of extensionality.[cite:@openstax2020calculus]

For some real number $b$ , if for every $x \in (b, a)$ , $f x = g x$ , then

$\lim_{x \to c} f x = \lim_{x \to c} g x .$

Squeeze theorem

See Squeeze theorem.