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# 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 $0<a-x<\delta$.

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
$0<x-a<\delta$.

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

$\lim_{{x\to 0^{-}}}H(x)=0\mbox{ and }\lim_{{x\to 0^{+}}}H(x)=1.$ |

## Mathematics Subject Classification

26A06*no label found*

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Broken by Mindspa ✓

Page images by Koro ✓

left-sided, left-handed by pahio ✓

down arrow notation by dimasad