# Understanding the formal definition of a limit

I started to write a short note about the limit and continuity some days ago. The question occured when I read some of the proofs, for example this <link> where the two limits were assumed to exist. Since they exist the formal definition of a limit would be applied and then using them to prove the goal. Now, the question is:

Let $f:A\to \mathbb{R}$ be a function. Do we say

$f(x)\to L$ as $x\to c$, if $$\forall \epsilon>0\exists\delta>0:\forall x\in A: 0<|x-c|<\delta \Longrightarrow |f(x)-L|<\epsilon$$

like many books say or this <link>, or do we say

$f(x)\to L$ as $x\to c$, if and only if $$\forall \epsilon>0\exists\delta>0:\forall x\in A: 0<|x-c|<\delta \Longrightarrow |f(x)-L|<\epsilon$$

like this <link>? I think the second one makes sense when it comes to the proofs.