Complex limit

The definition asks that \(|f(z)-L|\) be small whenever \(z\) lies in a small disk around \(z_0\), with \(z\neq z_0\). Since a disk contains points in every direction, the limit demands a single value regardless of path: in \(\mathbb{R}\) there are only two sides, in \(\mathbb{C}\) there are infinitely many directions. The value \(f(z_0)\) plays no role, and \(z_0\) must be an accumulation point of the domain.

The rules are the familiar ones: the limit of a sum, product and quotient (with nonzero limit in the denominator) is the sum, product and quotient of the limits. Moreover \(f=u+iv\) has limit \(L=a+ib\) if and only if \(u\to a\) and \(v\to b\) in the two-real-variable sense, which reduces the computation to limits in \(\mathbb{R}^2\).

To compute, simplify first: \((z^2+4)/(z-2i)=z+2i\) when \(z\neq 2i\), so the limit at \(2i\) is \(4i\). To show that a limit does not exist it suffices to give two paths with different values, as with \(\bar z/z\) at \(0\), which is \(1\) on the real axis and \(-1\) on the imaginary axis. The converse fails: agreeing along several paths does not prove existence.

A useful criterion: \(f(z)\to L\) is equivalent to \(|f(z)-L|\to 0\), and if \(|f(z)-L|\le C|z-z_0|\) the limit is \(L\). With the limit in hand, continuity is stated by saying that the limit equals the value of the function.