I get how for the limit as (x,y) approaches, say, (0,0), if there's a discontinuity along, say, x=0, then there won't be a limit, because no matter how small you make your $\delta$-circle, you won't get within $\epsilon$ of a limit. But I don't get how f could approach the same limit along every line through the point in question, but not approach the limit along another path. Could someone give an intuitive explanation (mathematical explanations are good too, but intuitive ones are preferred) of what a "path" means in terms of limits and how a limit could be the same within any $\delta$ along every line, but how the limit doesn't exist because it is different along a different non-linear path? If possible please include some mention of the $\delta$-circle.