When i first learned about limit i was taught that the limit $\lim \limits_{x \to a}$ f(x) = L exists if f(x) $\rightarrow$ L when x $\rightarrow$ a. Or that f(x) gets arbitrarily close to L when x gets arbitrarily close to a.
After leaning this intuitive but not rigorous definition, I looked at the delta epsilon definition. In the next paragraph i will try to relate the epsilon delta definition back to the first intuitive definition(ps: i'm not very familiar with mathy terms like "tolerance" or "error" etc, but i hope you understand me regardless.
The epsilon delta definition states, more or less, that if $\lim \limits_{x \to a}$ f(x) = L is true, then for every range $\epsilon$ around L, there exists some range $\delta$ around a such that all the points within $\delta$ of a evaluates to functions that are within $\epsilon$ of L. So if we set $\epsilon$ arbitrarily close to 0 , (1) then if we find a wide or big $\delta$ around a, it means that every function of those x'es within $\delta$ must be arbitrarily close to L, and that includes the x'es that are arbitrarily close to a. So the statement f(x) $\rightarrow$ L when x $\rightarrow$ a is true. (2) And if the $\delta$ is arbitrarily small when $\epsilon$ is arbitrarily small, it follows that f(x) $\rightarrow$ L when x $\rightarrow$ a
You're mostly right.
First, $\epsilon$ can get arbitrarily close to $0$, not $L$. In the definition we write
$$ \left | f(x) - L \right | < \epsilon $$
So $L$ does not have anything to do with the value of $\epsilon$ directly.
Also, there is no requirement for $\delta$ to be arbitrarily small. The best way to understand $\delta$ is to consider it as a function of $\epsilon$ (i.e. $\delta(\epsilon)$). You may (or may not) have a different $\delta$ for each $\epsilon$, and these may not get arbitrarily small; the only requirements are that it is positive, and that the function's value $f(x)$ stays within a distance of $\epsilon$ from $L$ as long as $x$ stays within a distance of $\delta$ from $a$.
For example given a constant function like $f(x) = 2$ (whose limit is obviously $2$ at $x = 0$ or any other point) $\delta$ can be any positive number for each $\epsilon$, since the function never "moves away" from $2$. So we could prove the limit by setting $\delta(\epsilon) = \frac{1}{\epsilon}$, which will get arbitrarily large as $\epsilon \to 0$!