Approaching from All Directions

Calculus 3 · Axiom Academy

Approaching from All Directions In two variables there are infinitely many ways to reach a point. A limit exists only if every path agrees — and one disagreement is enough to kill it. One point, infinitely many ways in In single-variable calculus a point has exactly two neighbors: you approach x=a from the left and from the right. In the plane, a point (a,b) can be approached along the axes, along any slanted line, along a parabola, a spiral — a whole infinity of paths. For a two-variable limit to exist, the function must head toward the same value no matter which path you take. Watch four crawlers close in on the origin along four different paths for . Each one carries the value f takes as it travels. The two axes settle on 0 — but the diagonal y=x settles on . Two paths, two answers. Checking only the axes would have "confirmed" a limit of 0 . The diagonal exposes the lie. Turn the dial: the answer follows the path Every straight line through the origin has the form y=mx . Drag the slope m and watch the value f approaches along that line. For it works out to — a number that moves as you turn the dial . If the destination depends on the road, there is no single limit. One family of paths — the lines — already contains a disagreement. That is all it takes. When the limit really is there

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