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# order of vanishing

Definition. Let $x_{0}$ be a zero of the real function $\Delta$. The order of vanishing of $\Delta$ at $x_{0}$ is $n$, if $\displaystyle\lim_{{x\to x_{0}}}\frac{\Delta(x)}{x^{n}}$ 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 $\Delta(h):=g(x_{0}+h)-f(x_{0}+h)$ of the ordinates has $n\!+\!1$-order of vanishing.

Related:

Multiplicity, OsculatingCurve

Synonym:

vanishing order

Type of Math Object:

Definition

Major Section:

Reference

Parent:

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## Mathematics Subject Classification

26E99*no label found*

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