How do you interpret those graphs they usually show about “precise definition of a limit”?
I assume you mean a graph like this one:
Let’s say that the limit of f(x) as x→a exists and is equal to L (as in the image).
What this means is that if you choose any little number ϵ, then there’s guaranteed to be another little number δ such that if you stay within a distance δ of a* – that is, if you stay inside the pink region – then f(x) is guaranteed to stay within a distance ϵ of L – that is, inside the yellow region.
*: When I say that you have to stay within a distance δ of a, a itself doesn’t count because the limit doesn’t say anything about what happens at a – only near a.
More informally, the limit exists if you can say that whenever x is any number really close to (but not necessarily at) a, then f(x) will be some number really close to L.