Definition of limit and axiom of choice In the definition of limit of a function ($\epsilon-\delta$ definition) we say certain statements such as for every $\epsilon>0$ there exist $\delta>0$ ....
Now my question is, is a choice function required to ensure that for every $\epsilon$ there exist a $\delta>0$? Moreover, how we can we test whether a mathematical statement depends on axiom of choice or not?
I really don't know whether this question has any meaning or not, so please help me.

The answer to your first question is: usually no. In $\epsilon-\delta$ questions, you usually have to construct a specific $\delta$ given $\epsilon$; i.e. your $\delta$ will be some function or equation in terms of $\epsilon$.
Having such a construction is exactly when you don't need the Axiom of Choice.  Very basically, you need the Axiom of Choice when you need to make an infinite number of choices that you cannot explicitly construct.
