The epsilon-delta definition of limit

Warning. This is a technical post.

We’ve all learnt about limits, but haven’t been formally introduced into it. The epsilon-delta definition is a standard formalisation of the concept of limits. I’ve seen our Maths teacher only use it once (and stop midway to try another way) in a limit that seemed unsolvable in any other way.

This goes into the very fundamentals of limits. It answers the basic question, “What is a limit?”

So now I’m going to quote Wikipedia below verbatim:

In calculus, the (ε, δ)-definition of limit is a formalisation of the notion of limit.

Let f(x) be a function defined on an open interval containing c (except possibly at c) and let L be a real number. Then we may make the statement

Lt_{x–>c} {f(x)} = L

if and only if:

If the value of x is within a specified δ units from c, this implies that f(x) is within a specified ε units from L.

Well, I know this just sounds unintelligible, but it’s only saying that if we restrict the function’s domain to any specified window, we can restrict the function’s range to a window that holds a definite relation with the window-to-the-domain for any windows chosen.

I don’t know if you understood that, but if you haven’t, then I recommend that you watch these series of videos (there are about four to six). I think you know Khan Academy…