One-sided limit

In calculus, a one-sided limit refers to either one of the two limits of a function ${\displaystyle f(x)}$ of a real variable ${\displaystyle x}$ as ${\displaystyle x}$ approaches a specified point either from the left or from the right.

The limit, as ${\displaystyle x}$ decreases in value approaching ${\displaystyle a}$ (${\displaystyle x}$ approaches ${\displaystyle a}$ "from the right" or "from above"), is denoted:

$${\displaystyle \lim _{x\to a^{+}}f(x)\quad {\text{ or }}\quad \lim _{x\,\downarrow \,a}\,f(x)\quad {\text{ or }}\quad \lim _{x\searrow a}\,f(x)\quad {\text{ or }}\quad f(a+).}$$

The limit, as ${\displaystyle x}$ increases in value approaching ${\displaystyle a}$ (${\displaystyle x}$ approaches ${\displaystyle a}$ "from the left" or "from below"), is denoted:

$${\displaystyle \lim _{x\to a^{-}}f(x)\quad {\text{ or }}\quad \lim _{x\,\uparrow \,a}\,f(x)\quad {\text{ or }}\quad \lim _{x\nearrow a}\,f(x)\quad {\text{ or }}\quad f(a-).}$$

If the limits from the left and right both exist and are equal, then the limit of ${\displaystyle f(x)}$ as ${\displaystyle x}$ approaches ${\displaystyle a}$ exists. Conversely, if the limit of ${\displaystyle f(x)}$ as ${\displaystyle x}$ approaches ${\displaystyle a}$ exists, then the limits from left and right both exist and are equal. Consequently, the limit as ${\displaystyle x}$ approaches ${\displaystyle a}$ is sometimes called a "two-sided limit". It is denoted: $${\displaystyle \lim _{x\to a}f(x).}$$

In some cases in which the two-sided limit does not exist, the two individual one-sided limits nonetheless exist and they are then necessarily unequal.

It is possible for only one of the two one-sided limits to exist. It is also possible for neither of the two one-sided limits to exist.