# order of vanishing

Let $x_{0}$ be a zero (http://planetmath.org/ZeroOfAFunction) 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.

Title order of vanishing OrderOfVanishing 2013-03-22 17:57:15 2013-03-22 17:57:15 pahio (2872) pahio (2872) 6 pahio (2872) Definition msc 26E99 vanishing order Multiplicity OsculatingCurve