In this section on wikipedia for the epsilon-delta definition of the limit, we are given a logical statement. https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit#Precise_statement_for_real_valued_functions
$$\lim_{x \to c} f(x) = L \iff (\forall \epsilon > 0, \exists \delta > 0, \forall x \in D, 0 < |x-c| < \delta \implies |f(x) - L| < \epsilon)$$
Now, there are a few things I have to ask about this statement of logic.
First of all, I presume that the limit is defined to exist (with some unknown value) if the following is true:
$$\exists L,\forall \epsilon > 0, \exists \delta > 0, \forall x \in D, 0 < |x-c| < \delta \implies |f(x) - L| < \epsilon$$
Am I wrong? If so, is there such a statement?
Secondly, I presume that L is the value of the limit and that it is unique. I mean I know the limit is unique but is this logical statement implying a uniqueness for L? That is to say, can only one L fulfill it and is it the limit value?
Finally, how would I rewrite it so that it gives the value of the left or right hand limits? Would I just get rid of the absolute values? I'm not sure.
The reason I am asking these things is that I believe I can (mostly) rewrite predicate logic statements as indicator functions involving infinite sums and products and I wish to make sure I have my logic correct before (attempting) to express the limit itself in such a manner. Hence, I'm asking some pretty finicky questions to make sure I understand properly.
I've never messed with the epsilon-delta definition before so I apologize if any of these are somewhat trivial.
First part, you are absolutely correct. Just add the quantifier $\exists L$ to define the notion of the limit existing.
Second, you can prove this explicitly. Suppose $\lim_{x\to c}f(x)=L$ and $\lim_{x\to c}f(x)=L'$ with $L<L'$. You can find a contradiction using the definition of the limit (hint: choose $\epsilon=\frac{L'-L}{2}$).
For the last question, we define $\lim_{x\to c^+}f(x)=L$ iff $$\forall\epsilon>0,\exists\delta>0,0<x-c<\delta\implies |f(x)-L|<\epsilon$$ Notice that $0<x-c<\delta\iff c<x<c+\delta$, which is the range we expect for a limit from above. With this in mind, can you figure out how we would write $\lim_{x\to c^-}f(x)$?
Let me know if I can clarify anything!