In the epsilon-delta definition, what is wrong if I said: "given delta, there exists an epsilon"? WHY are we always given $\epsilon > 0$ first, then solving for a $\delta>0$? This is in the limit definition.
I want to ask:

Can we say "given $\delta>0$, there exists $\epsilon>0$"? Since we can always solve for one given the other.

I found three counterexamples, but I don't understand them:


*

*Let $f(x) = \sin x$, let $L$ and $\delta$ be arbitrary real numbers.  Then $\epsilon = |L| + 2$ satisfies your definition. (from post)  
Q: What's wrong with setting $\epsilon = |L| + 2$? It's big, but it's not wrong! 

*Let $f(x) = 1/x$, and let $a = 1$.  The definition fails for  $\delta \ge 1$, since for any $\epsilon$ we can choose $x=1/(L+\epsilon)$ if $L+\epsilon > 1$, so that $f(x)-L \ge \epsilon$. (from post)
Q: What are they saying here? At $x=1$, the definition fails for $\epsilon \ge 1$ too! The problem is not $\delta$. The problem is the function is undefined for $x \le 0$.

*Counterexample: $\lim\limits_{x \to 0} f(x) = L$ 
$f(x) = \begin{cases} \sin \frac{1}{x}, & x \ne 0 \\ 0, & x = 0 \end{cases}$ 
Given any $\delta > 0$, we can  find $\epsilon > 0$ such that $|f(x) - L| < \epsilon$ whenever $|x| < \delta$.  For instance, set $\epsilon = 2$; then any choice of $L \in (-1,1)$ will satisfy this "reversed" situation. (from post)
Q: I don't see how setting $\epsilon = 2$ violates any definition. I mean, we did find a $\epsilon$ for a given $\delta$.
Thanks all for the pouring answers, I'll get back to each one personally. If I did not choose an answer, that means all submissions are still welcomed! The best answer will be chosen based on # of upvotes (50%) and if I understood it and agree it's the best (50%).

Because $\epsilon$ can be unbounded. Let's say we choose $\delta=3$., then is $\epsilon = 6$. But, does it have to be 6? No! It can be 7, 8, 9, 1000. In fact, any number bigger than 6 will do. So, choosing $\delta$ first leaves $\epsilon$ unbounded.

Choosing $\epsilon$ first is like reverse engineering. To me, changing $y$ first, and watch what happens to $x$ is weird. Because I'm used to x as the independent variable, and y as the dependent variable.

Q: Why does "reverse engineering" work? What's the intuition behind it?