Limits and Continuity in the Complex Plane: A Clear Guide

The limit of f at z_0 is L if f(z) gets arbitrarily close to L as z gets close to z_0: for every ε > 0 some δ > 0 gives |z − z_0| < δ ⇒ |f(z) − L| < ε. Unlike real calculus, z approaches from every direction in the plane, so the limit must agree along all paths or it does not exist. Continuity means lim_{z→z_0} f(z) = f(z_0).

Limits and continuity found complex differentiation and integration: every theorem about analytic functions assumes continuity, and path-independence of integrals depends on it.

Key Concepts

• ε–δ with discs: neighborhoods are discs |z − z_0| < δ, not intervals.
• Path test: two paths with different limiting values prove the limit does not exist.
• Component criterion: the complex limit exists exactly when u → Re(L) and v → Im(L).
• Continuity via components: f = u + iv is continuous exactly when u and v are.
• Algebra of limits: sums, products, and quotients (nonzero denominator limit) of continuous functions stay continuous.

How It Works

1. To prove a limit exists, bound |f(z) − L| by something tending to zero, using |Re z|, |Im z| ≤ |z| and the triangle inequality.
2. To disprove, find two paths disagreeing. Classic: f(z) = z̄/z at 0.
3. Along the real axis z = x ≠ 0: z̄/z = 1.
4. Along the imaginary axis z = iy: z̄/z = −1. Since 1 ≠ −1, lim_{z→0} z̄/z does not exist.

Worked Example

Evaluate lim_{z→i} (z² + 1)/(z − i). Substitution gives 0/0; factor: z² + 1 = (z − i)(z + i), so for z ≠ i the quotient is z + i → 2i. Numeric check at z = i + 0.01: z² + 1 = (−1 + 0.02i + 0.0001) + 1 = 0.0001 + 0.02i; dividing by 0.01 gives 0.01 + 2i, within 0.01 of 2i. The function is undefined at z = i, but the limit exists: a removable discontinuity.

Common Misconceptions

• "Checking both axes suffices." A function can agree on the axes yet fail along a diagonal; every path must agree.
• "z̄ is discontinuous since it is not differentiable." Conjugation is continuous everywhere (|z̄ − w̄| = |z − w|); differentiability is the stricter demand.
• "0/0 means no limit." It only means more work; factoring often reveals a clean limit.

Real-World Applications

• Continuity makes numerical evaluation trustworthy: small input errors give small output errors.
• Control theory leans on continuity of transfer functions for stability arguments.
• Contour integration requires the integrand to be continuous along the path.

Complex limits demand agreement from all directions, making the path test the fastest way to expose failure. Continuity then behaves familiarly and quietly carries differentiation and integration on its back.