limit of a function of several variables

In one variable, approaching a point means coming in from the left or from the right — just two roads. For a function f(x, y) of two variables, the point (a, b) sits in a plane, and you can walk toward it from infinitely many directions, along straight lines, spirals, parabolas, any curve at all. The limit of f as (x, y) approaches (a, b) is the single number L that f gets close to no matter which of those infinitely many roads you take in.

Precisely, lim of f(x, y) as (x, y) -> (a, b) equals L when f(x, y) can be forced as close to L as you like simply by keeping the point (x, y) close enough to (a, b) — close measured by the distance sqrt((x - a)^2 + (y - b)^2). The key word is every approach. If two different paths into (a, b) give two different target values, the limit does not exist, full stop. This is why the everyday tricks — plug in, or take the limit in x first and then in y — can quietly lie: an iterated limit can exist while the true two-dimensional limit does not.

This subtlety is not academic. Every later idea in multivariable calculus — continuity, the partial and total derivative, differentiability — is built on this limit, so getting it wrong poisons everything downstream. In practice you confirm a limit exists by switching to polar coordinates (let r -> 0 and check the answer is independent of the angle), and you disprove one by finding two paths, often y = mx versus y = x^2, that disagree.