Approaching from All Directions

Complex Analysis · Axiom Academy

Approaching from All Directions On the real line a limit has two ways in. In the complex plane it has infinitely many — and they all have to agree. A limit you have to defend from every angle In ordinary calculus, asks just two questions: what happens as x slides in from the left, and from the right. A complex number z doesn't live on a line — it lives in a plane , so it can march toward a target point z_0 from infinitely many directions, or wind in along any curving path at all. Watch eight markers set off toward z_0 = i from completely different directions, all feeding the same rule f(z) = z^2 + (1-2i)z . Their outputs are computed live as they travel — and no matter which way they came in, they pour into the same landing point L . When that happens for every path, the limit exists . Eight directions in, one value out. The outputs start scattered, then collapse onto a single L — that collapse is exactly what "the limit exists" means. One function that can't make up its mind Here is the classic troublemaker: , where is the complex conjugate. Drag the dial to choose the direction you approach z_0 = 0 from. The value f(z) along that ray is computed for you. Slide to the real axis and to the imaginary axis, and watch the answer flip. Real axis says 1 , imaginary axis says -1 — the directions disagree, so the limit does not exist. Two paths, two different answers. A single bad pair of directions is enough to kill the limit. A function that agrees with itself

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