Epsilon-delta definition

The epsilon-delta definition of a limit says that the limit below exists

$$\underset{z\to z_0}{\lim }g(z)=l$$

if we can find an $ϵ>0$ where there exists a $\delta >0$ such that $|z-z_0|<\delta$, then $|g(z)-l|<ϵ$. This definition forces $g(z)$ to approach $l$ no matter how $z$ approaches $z_0$, which is especially useful in the multivariable definition of the limit.

This is a fairly difficult proof in introductory real analysis classes, so engineering classes don’t really look at this in depth.

Intuitive understanding

The above looks like a sorcerer spell, so Prof Nachman explained it in terms of an analogy. Say we’re in a machine shop (or some other precision-based place) where we want an error of $|g(z)-l|<0.001$. What we’re essentially describing is a tolerance. The $ϵ$ is an error, and $\delta$ is a distance.

When we discuss the epsilon-delta definition, we aim to generalise this for some arbitrarily small tolerance, and we want to get as close to $l\forall z$ sufficiently close to $z_0$.