2D Limits

1D Limit

We say

$$\underset{x\to x_0}{\lim }f(x)=L$$

exactly when for every $ϵ>0$, there is $\delta >0$ such that whenever $\mid x-x_0\mid <\delta$ it follows that $\mid f(x)-L\mid <ϵ$.

Recall: for the limit to exist, the left and right hand limits must exist and be the same.

2D Limit

We say

$$\underset{x,y\to x_0,y_0}{\lim }f(x,y)=L$$

exactly when for every $ϵ>0$, there is a $\delta >0$ such that whenever $\sqrt{(x-x_0{)}^2+(y-y_0{)}^2}<\delta$ and $(x,y)$ is in the domain of $f$, it follows that $\mid f(x)-L<ϵ\mid$.