# one-sided limit

Definition Let $f$ be a real-valued function defined on $S\subseteq\mathbb{R}$. The left-hand one-sided limit at $a\in\mathbb{R}$ is defined to be the real number $L^{-}$ such that for every $\epsilon>0$ there exists a $\delta>0$ such that $|f(x)-L^{-}|<\epsilon$ whenever $0.

Analogously, the right-hand one-sided limit at $a\in\mathbb{R}$ is the real number $L^{+}$ such that for every $\epsilon>0$ there exists a $\delta>0$ such that $|f(x)-L^{+}|<\epsilon$ whenever $0.

Common notations for the one-sided limits are

 $\displaystyle L^{+}$ $\displaystyle=$ $\displaystyle f(x+)=\lim_{x\to a^{+}}f(x)=\lim_{x\searrow a}f(x),$ $\displaystyle L^{-}$ $\displaystyle=$ $\displaystyle f(x-)=\lim_{x\to a^{-}}f(x)=\lim_{x\nearrow a}f(x).$

Sometimes, left-handed limits are referred to as limits from below while right-handed limits are from above.

Theorem The ordinary limit of a function exists at a point if and only if both one-sided limits exist at this point and are equal (to the ordinary limit).

Example The Heaviside unit step function, sometimes colloquially referred to as the diving board function, defined by

 $H(x)=\begin{cases}0&\mbox{ if }~{}x<0\\ 1&\mbox{ if }~{}x\geq 0\end{cases}$

has the simplest kind of discontinuity at $x=0$, a jump discontinuity. Its ordinary limit does not exist at this point, but the one-sided limits do exist, and are

 $\lim_{x\to 0^{-}}H(x)=0\mbox{ and }\lim_{x\to 0^{+}}H(x)=1.$
 Title one-sided limit Canonical name OnesidedLimit Date of creation 2013-03-22 12:40:28 Last modified on 2013-03-22 12:40:28 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 11 Author CWoo (3771) Entry type Definition Classification msc 26A06 Synonym limit from below Synonym limit from above Synonym left-sided limit Synonym left-handed limit Synonym right-sided limit Synonym right-handed limit Related topic Limit Related topic OneSidedDerivatives Related topic IntegratingTanXOver0fracpi2 Related topic OneSidedContinuity Defines Heaviside unit step function