I'm always confused when it comes to quantifiers (forall and exist)
let's take the following examples I would like if someone can explain me how to implement intuition so I can visualize what is actually happening so I can see it clearly and understand why it's wrong.
Thanks in advance!
If you are trying to express the claim that $\lim_{x \to 2} f(x) = 4$, then the correct statement is
$$ \forall \varepsilon > 0 ~ \exists \delta > 0 ~\forall x \in \mathbb{R} ~ \big(|x - 2| < \delta \rightarrow |f(x) - 4| < \varepsilon\big) \enspace. $$
Why? Because the order in which the quantifiers appear in your sentence is the order in which choices are made.
If you read the claim as, "No matter how close you need $f(x)$ to be to $4$, you can pick $x$ close enough to $2$ to meet your goal," you see that $\varepsilon$, the distance of $f(x)$ from $4$, must be chosen before $\delta$, the distance from $x$ to $2$.
It is often helpful to think of sentences with quantifiers as specifying a game between two players. One player, A, chooses the universally quantified variables, while the other player, E, chooses the existentially quantified variables. The "universal player" A tries to make the sentence false; the "existential player" E tries to make it true. The quantified sentence is true if E has a strategy to win the game.
In our case, A picks $\varepsilon$. If indeed $\lim_{x \to 2} f(x) = 4$, then E can find $\delta$ such that no matter which $x$ A chooses next, either $|x-2| \geq \delta$ or $|f(x) - 4| < \varepsilon$.