Eli5: What does epsilon and delta have to do with finding limits?

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The value of either of these is just going to be a function of the other. What does delta and epsilon even mean? What do they establish?

In: Mathematics
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Epsilon and delta and just dumbie variables. Inherently then don’t mean anything it’s your job to find out what they are if the question asks for it. The names and symbols are just used universally and usually when they are introduced mathematicians know what dance they have to do to solve the proof.

Epsilon refers to the y-tolerance and delta refers to the x-tolerance. For example, if you’re trying to find the limit of a function between two x values (denoted as “Delta”) then the limit is said to reside between two y values (denoted as “Epsilon”).

This is going to be a bit hard to explain without maths notation, and Reddit isn’t great for that, but let’s give it a go.

Start with [this page](https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit#Precise_statement_for_real-valued_functions).

> Let *f* be a real-valued function defined on a subset *D* of the real numbers. Let *c* be a limit point of *D* and let *L* be a real number.

Putting that into ELi5ish terms, we have some function f(x), defined for x in D, and we want to see if it has a limit L when x = c. I.e. lim x->c f(x) = L

> We say that lim x→c f(x) = L if for every ε > 0 there exists a δ > 0 such that, for all x ∈ D if 0 <|x−c|< δ, then |f(x)−L|<ε.

So what we are doing here is a kind of challenge. You give me an ε as a target. Given your ε, I have to find a δ so that if our x-values are within δ of c, then f(x) is within ε of L.

The ε is the challenge you give me, the δ is the reply I give back. And if I can find a δ for any ε you give me, I win and the function has a limit.

[This diagram may help a bit](https://en.wikipedia.org/wiki/File:L%C3%ADmite_01.svg). This is for continuity but the same principle applies. You give me an ε. For the function to be continuous I have to find some δ so that while we are within δ of c, f(x) is always within ε of L. No matter how small you make ε, no matter how narrow you make the target area, if I can find a δ so f(x) hits the target area, I win and the function is continuous there. I can find a region around the point where the function is arbitrarily close to L.