# order of vanishing

Definition. Let ${x}_{0}$ be a zero (http://planetmath.org/ZeroOfAFunction) of the real function $\mathrm{\Delta}$. The order of vanishing of $\mathrm{\Delta}$ at ${x}_{0}$ is $n$, if $\underset{x\to {x}_{0}}{lim}{\displaystyle \frac{\mathrm{\Delta}(x)}{{x}^{n}}}$ has a non-zero finite value.

Usually, ${x}_{0}$ of the definition is 0.

Example. If the curves $y=f(x)$ and $y=g(x)$ have in the point $({x}_{0},{y}_{0})$ the order of contact $n$, then the difference^{} $\mathrm{\Delta}(h):=g({x}_{0}+h)-f({x}_{0}+h)$ of the ordinates has $n+1$-order of vanishing.

Title | order of vanishing |
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Canonical name | OrderOfVanishing |

Date of creation | 2013-03-22 17:57:15 |

Last modified on | 2013-03-22 17:57:15 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 6 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 26E99 |

Synonym | vanishing order |

Related topic | Multiplicity |

Related topic | OsculatingCurve |