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
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