## Limits

Tonight, I was reading about the fundamental definition of limits of functions, and for the first time, really understood it.  The context was a limit in multiple variables, as

$\displaystyle\lim_{(x,y)\to (0,0)}f(x,y) = L$

The interesting issue is that in two dimensions like this, showing the limit exists is harder than one dimensions, since there are infinitely many ways to approach (a,b).   The definition for continuity becomes, $f(x,y)$ is continuous at the point $(a,b)$ if the limit exists and is $f(a,b)$, or

$\displaystyle\lim_{(x,y)\to (a,b)}f(x,y) = f(a,b)$

Somehow, I never fully understood the 1D case until I saw the explaination and example in 2D.