Definition:Limit of Real Function/Intuition

Definition

Though the founders of Calculus viewed the limit:

$\ds \lim_{x \mathop \to c} \map f x$

as the behavior of $f$ as it gets infinitely close to $x = c$, the real number system as defined in modern mathematics does not allow for the existence of infinitely small distances.

But:

$\exists L: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: 0 < \size {x - c} < \delta \implies \size {\map f x - L} < \epsilon$

can be interpreted this way:

You want to get very close to the value $c$ on the $\map f x$ axis.
This degree of closeness is the positive real number $\epsilon$.
If the limit exists, I can guarantee you that I can give you a value on the $x$ axis that will satisfy your request.
This value on the $x$ axis is the positive real number $\delta$.