The epsilon-delta definition for limits states that (from Wikipedia) for all real $\epsilon > 0$ there exists a real $\delta > 0$ such that for all $x$ with $ 0 < |x − c | < \delta$, we have $|f(x) − L| < \epsilon$ - however, the definition of the limit requires only the existence of some $\delta>0$ for any $\epsilon>0$. The part I am having trouble understanding is why there are no details as to the "intuitive" decrease of the δ as ε grows smaller.
I realize that saying that as ε approaches zero δ also approaches zero would use the non-rigorous intuition of a limit in a definition meant to make the limit a rigorous part of mathematics, but why is it unnecessary to show the relationship between epsilon and delta besides the proof of existence? Is there some other implication of a function that I am missing that is the reason only the proof of existence is in this definition?
EDIT: If we consider the dependence of on an epsilon on a decreasing delta, can the limit exist if epsilon is increasing as delta decreases? If so, why?