# 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 $\u03f5>0$ there
exists a $\delta >0$ such that $$ whenever $$.

Analogously, the *right-hand one-sided limit* at $a\in \mathbb{R}$ is the
real number ${L}^{+}$ such that
for every $\u03f5>0$ there exists a $\delta >0$ such that $$ whenever
$$.

Common notations for the one-sided limits are

${L}^{+}$ | $=$ | $f(x+)=\underset{x\to {a}^{+}}{lim}f(x)=\underset{x\searrow a}{lim}f(x),$ | ||

${L}^{-}$ | $=$ | $f(x-)=\underset{x\to {a}^{-}}{lim}f(x)=\underset{x\nearrow a}{lim}f(x).$ |

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

$$ |

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

$$\underset{x\to {0}^{-}}{lim}H(x)=0\text{and}\underset{x\to {0}^{+}}{lim}H(x)=1.$$ |

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 |