independence of the limit from the direction of approach

This is the single feature that makes complex limits demanding, so it deserves a name of its own. For the limit of f(z) as z tends to z_0 to exist and equal L, the output must approach L no matter how z creeps toward z_0 — and in the plane there are infinitely many ways to creep in: straight along any of the infinitely many directions, along a curve, along a spiral. Every one of these approaches must agree on the same L.

This gives you a clean and powerful test for non-existence. If you can find two different paths into z_0 along which f tends to two different values, the limit cannot exist — done. The classic case is f(z) = z-bar/z near z_0 = 0. Approach along the positive real axis (z = t, t -> 0+): then z-bar/z = t/t = 1. Approach along the imaginary axis (z = i t): then z-bar/z = (-i t)/(i t) = -1. Two routes, two answers, so no limit. Notice you did not even have to test every path — one disagreement is fatal.

Contrast this with the single-variable real limit, where 'all approaches' meant only the left limit and the right limit; agreeing on two numbers was enough. The leap to a full disk of directions is why a complex limit, and the complex derivative built on it, is so much more rigid than its real cousin. That rigidity is not a nuisance — it is the source of the astonishing rigidity and smoothness of holomorphic functions later on.