limit (multiple dimensions)

Assume we have a function with multiple variables, \[f(x,y)\], and would like to describe its limit as the point \[(x,y)\] approaches a particular point \[\left( x_{0},y_{0} \right)\]. This is written as \[\lim_{\left( x,y \right)\to \left( x_{0},y_{0} \right)}f(x,y)=L\].

In this case, we need to change the inequality \[0<\left| x-a \right|<\delta\] in the definition. Since we know that the shortest distance between two points would be \[\sqrt{\left( x-x_{0} \right)^{2}+\left( y-y_{0} \right)^{2}}\], there we can change the definition as follows:
\[\lim_{\left( x,y \right)\to \left( x_{0},y_{0} \right)}f(x,y)=L\] if for every \[\epsilon>0\], there is a \[\delta>0\] such that for all \[\left( x,y \right)\] in the domain of \[f\] that satisfy \[0<\sqrt{\left( x-x_{0} \right)^{2}+\left( y-y_{0} \right)^{2}}<\delta\], the inequality \[\left| f(x)-L \right|<\epsilon\] holds.