path dependence of limits
Imagine standing at a point in a flat plane and walking toward a target spot from every conceivable direction, watching the function's height as you arrive. If every walker reports the same arrival height, that common value is the limit. Path dependence is what happens when the walkers disagree — different routes into the same point yield different limiting values — and it is the single most common reason a multivariable limit fails to exist.
The path test exploits this directly. To approach (0, 0), substitute a family of curves through the origin, say y = m x, and compute the limit in the single remaining variable. If the answer depends on m, you are done: the limit cannot exist, because picking different slopes m gives different values. Straight lines do not always expose the problem, so the next weapons are parabolas y = k x^2 (which can disagree with all the lines) and, for a clean all-directions sweep, polar coordinates x = r cos(theta), y = r sin(theta) with r -> 0: if the result still contains theta, the limit is path-dependent.
This is the practical engine behind why multivariable limits are subtle. A path that gives a finite, consistent-looking answer never proves existence — it only fails to disprove it. The asymmetry is total: any single bad pair of paths kills the limit, but no finite list of good paths can save it. That is why establishing existence ultimately needs polar coordinates or an epsilon-delta bound, while disproving existence is the quick, satisfying part of the job.
For f(x, y) = x^2 y / (x^4 + y^2), every straight line y = m x gives limit 0, which looks reassuring — but the parabola y = x^2 gives 1/2. The lines all agreed and were all wrong; the parabola exposes the path dependence.
When all straight lines agree, try parabolas — a curve tuned to the function's powers can still break the limit.
Agreement along infinitely many straight lines is still not existence. The classic counterexample x^2 y / (x^4 + y^2) gives 0 on every line yet 1/2 on a parabola — proof that the path test can only disprove, never confirm.