Limit of complex function

The limit of a complex function $f$ at a point $z_0$ is defined exactly like a real limit, but with absolute values interpreted as complex moduli:

${\lim }_{z\to z_0}f(z)=L$

means: for every $\epsilon >0$ there is a $\delta >0$ such that $0<|z-z_0|<\delta$ implies $|f(z)-L|<\epsilon$.

What’s subtly different from the real case

”$z$ close to $z_0$” means anywhere in a 2D disk of radius $\delta$ around $z_0$. The complex limit must agree along every direction of approach — along the real axis, the imaginary axis, any diagonal, any spiral. If two approach paths give different limiting values, the limit doesn’t exist.

This is a much more demanding condition than the one-real-variable case, where you only have to check the left limit and the right limit.

A diagnostic failure

$\underset{z\to 0}{\lim }\frac{z}{|z|}$.

Approach along the positive real axis ($z=x,x\to 0^{+}$): $\frac{x}{|x|}=1$.

Approach along the positive imaginary axis ($z=iy,y\to 0^{+}$): $\frac{iy}{|iy|}=\frac{iy}{y}=i$.

Different directional limits, so the limit doesn’t exist.

This kind of “approach-dependent” failure is the engine of why most functions ($\bar{z}$, $\mathrm{Re}(z)$, $|z|$) fail the Complex derivative test.

A succeeding limit

$\underset{z\to z_0}{\lim }{z}^2=z_0^2$.

Setting $z=z_0+h$ with $h\to 0$,

$z^2=z_0^2+2z_{0}h+h^2\to z_0^2$

along every direction. The limit exists and equals $z_0^2$. This is the foundation for $z^2$ being differentiable.

Standard rules

Sum, product, quotient (when the denominator’s limit is nonzero), and composition rules carry over from real-variable theory. The proofs are identical with $|\cdot |$ reinterpreted.

Reduction to real-variable limits

Writing $f(z)=u(x,y)+iv(x,y)$ and $L=a+ib$,

${\lim }_{z\to z_0}f(z)=L⟺{\lim }_{(x,y)\to (x_0,y_0)}u(x,y)=a\text{ and }{\lim }_{(x,y)\to (x_0,y_0)}v(x,y)=b.$

So a complex limit is two simultaneous real two-variable limits — both of which must agree along every path of approach.

In context

The complex limit is the foundation for:

• Continuity of complex functions.
• The Complex derivative (difference quotient as a complex limit).
• Convergence of sequences and series in $\mathbb{C}$.

The “every direction must agree” condition is what makes complex analysis rigid in a way real analysis is not. A function differentiable once on an open region turns out to be infinitely differentiable, equal to its own Taylor series, and constrained globally by its values on any small piece. All of this rests, ultimately, on the strict definition of the complex limit.