I do have an issue understanding this definition (I know there have been multiple questions around this, but I couldn't find a direct answer to my question, so please bear with me).
The informal definition of a limit states that we need to find a number $L$, the value of the limit, such that when $x$ approaches some $a$, $f(x)$ approaches $L$.
The epsilon-delta definition states that for all $\epsilon>0$ there exists a $\delta>0$ such that, whenever $|x−a|<\delta$ then $|f(x)−L|<ϵ$.
What I do not understand is how the "$x$ approaches $a$ implies $f(x)$ approaches $L$" is implied. Intuitively, for arbitrary small $\epsilon, \delta$ should get arbitrary small too, according to the intuitive definition.
But I do not see any reason why, as $\epsilon$ gets small, $\delta$ cannot get larger.