the limit of a complex function
Saying f(z) approaches a value L as z approaches a point z_0 means the same intuitive thing as in ordinary calculus: you can make the output f(z) as close to L as you like by taking the input z close enough to z_0. We write the limit of f(z) as z tends to z_0 equals L. 'Close' is measured by the modulus: |f(z) - L| small means the output is near L, and |z - z_0| small means the input is near z_0.
Precisely: the limit is L if, for every tolerance epsilon > 0, there is a radius delta > 0 such that whenever 0 < |z - z_0| < delta we have |f(z) - L| < epsilon. The picture is two disks: given any small target disk of radius epsilon around L in the w-plane, you can find a small disk of radius delta around z_0 in the z-plane whose points (except z_0 itself) all land inside the target. The value of f at z_0 is deliberately ignored — only the approach matters.
The catch, and the reason complex limits are stricter than real ones, is that |z - z_0| < delta is a whole disk: z may approach z_0 from any direction in the plane — along the real axis, along the imaginary axis, along a spiral — and the same single L must be reached on every route. On the real line you only had two directions (left and right) to reconcile; in the plane you have infinitely many. If two approaches give different answers, the limit simply does not exist.
The limit of z^2 as z tends to i is -1: as z gets close to i from any direction, z^2 gets close to i^2 = -1. By contrast, the limit of z-bar/z as z tends to 0 does not exist, because approaching along the real axis gives 1 while approaching along the imaginary axis gives -1.
A complex limit exists only if every direction of approach yields the same value.
A handy shortcut: the limit of f = u + i v exists and equals A + i B exactly when u(x, y) tends to A and v(x, y) tends to B as (x, y) tends to (x_0, y_0). A complex limit is just a pair of ordinary two-variable real limits.