The precise definition of a limit is something we use as a proof for the existence of a limit.
Let’s start by stating that f(x) is a function on an open interval that contains x=a but that the function doesn’t necessarily exist at x=ax=a. We can say that
lim x → a f (x) = L
if for every number ϵ > 0 there is some number
δ > 0 such that
\ f (x) – L\ < ϵ whenever 0 < \ x – a \ < δ