Multivariate calculus from first principles
On a line you could only sneak up on a point from two sides, left and right. In the plane and beyond, you can approach a point from infinitely many directions, along any path you like. That extra freedom makes limits in Rⁿ genuinely harder, and this lesson is a warning more than a recipe.
A function f has limit L at a point p only if it heads toward the same L no matter which path you take in. If two different paths give two different answers, the limit simply does not exist.
You agree to meet a friend at a fountain in the middle of a plaza. You can walk toward it from the north entrance, the east alley, or any winding diagonal across the square, but you must end up at the same fountain. A limit in Rⁿ demands exactly this: the function must head for one value no matter which path you take in. If two approaches disagree on where they land, there is no meeting spot, and the limit does not exist.