91Ó°ÊÓ

The goal is to provide a definition for $\underset{(x,y)→(a,b)}{\lim }f(x,y)=∞$.

The infinite limit at a point, represented as $\underset{\left(x\right)→\left(a\right)}{\lim }f\left(x\right)=∞$, denotes the existence of an infinite limit for every $M>0$, such that if $x∈(a-\mathrm{δ},a)∪(a,a+\mathrm{δ})$, then $f\left(x\right)∈(M,∞)$

The limit of a two-variable function fis L, expressed as $\underset{\left(x\right)→\left(a\right)}{\lim }f\left(x\right)=L$if, for every $\mathrm{Î\mu }>0$, there is $\mathrm{δ}>0$such that $\left|f\left(x\right)-L\right|<\mathrm{Î\mu }$whenever role="math" localid="1653486855833" $0<\left|\left|x-a\right|\right|<\mathrm{δ}$.

To define the infinite limit of a function in two variables, combine the two definitions.

The infinite limit of a function in two variables at a point, represented as$\underset{(x,y)→(a,b)}{\lim }f(x,y)=∞$, indicates that for any $M>0$, there exists $\mathrm{δ}>0$ such that if is $0<\sqrt{{(x-a)}^2+{(y-b)}^2}<\mathrm{δ}$, then $f\left(x\right)∈(M,∞)$.