Epsilon-Delta for Complex Limits

Complex Analysis · Axiom Academy

LESSON Epsilon-Delta for Complex Limits The same ε-δ promise as on the real line — but now ε and δ control disks in the complex plane, not intervals. 1. The Definition — a Disk Inside a Disk We say when, no matter how tight a target tolerance you demand, there is a closeness that delivers it: read: "inside the punctured δ-disk around z_0 , f stays inside the ε-disk around L " Geometrically it is a promise about two disks . Draw the ε-disk of radius around the target L in the output plane. The definition guarantees a δ-disk of radius around z_0 in the input plane whose entire image lands inside that ε-disk . The little hole " " just excludes z_0 itself — the limit never asks about the value at z_0 . 2. Tighten ε, and δ Must Tighten Too The limit exists only if the disk guarantee survives every choice of — including ones approaching 0 . So is not a fixed number; it is a response to . Demand a smaller ε-disk and you must answer with a smaller δ-disk to keep the image inside. For each you produce a . As , your — yet for every positive a working exists. The challenger picks — the radius of the target disk around L . Nothing is off-limits; it can be tiny. You respond with — small enough that the δ-disk's image is swallowed by the ε-disk. 3. How to Find δ: Work Backwards from the Goal Every ε-δ proof has the same shape. You start at the finish line — the quantity |f(z)-L| — and squeeze it until |z-z_0| appears. Whatever multiplies |z-z_0| tells you how to choose .

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