Functional Relationship Between Epsilon and Delta

Question

In my textbook on vector calculus, I was studying up on the epsilon-delta definition of limits (new to me at the time). I believe I've understood what it's all about intuitively, and appreciate the rigour of it; however there was a statement (more of side note) that said something along the lines of "generally you must find a functional relationship between epsilon and delta". I don't have the book handy, and can't recall if it used the word "generally" or "usually", hence I'm now wondering if a functional relationship is necessary, or if there are other, "sneaky" ways to prove a limit converges using epsilon-delta. I appreciate any insight into the question. I remember reading a second definition that talked about a "neighborhood" around a point, but couldn't see the rigour behind it.

Answer 1

What you have to do is to show that for any $\varepsilon>0$, there is at least one $\delta>0$ which works. Theoretically, there is no need to supply one, only show that one exists.

However, in practice, given $\varepsilon$, actually supplying a $\delta$ that works turns out to be the most practical way to achieve this. So that's what we do not of the time: for any $\varepsilon>0$ we supply a single, concrete $\delta$. That happens to be exactly what a function is.

Answer 2

For a given $\epsilon >0$ you need to find a $\delta$ such that whenever $0<|x-a|<\delta$ we get $0\le |f(x)-L|<\epsilon $

As you see, we are given an $\epsilon$ and we are trying to find a $\delta$ which may or may not depend on $\epsilon$ which makes a relation hold.

Generally $\delta$ is a function of $\epsilon$ for example you may get $\epsilon = \delta/5$ but do not worry if you find a $\delta$ independent of $\epsilon$ as long as it works for you.