Definition of Limit: CCD.DL.
  Consider a function $f$ and a number $a$ in an open interval, $O$, for which all $x \ne a, x \in O$ are in the domain of $f$. The number $L$ is called the limit of $f$ as $x$ approaches $a$, ($ x \to a$), denoted $\lim_{x \to a} f(x) = L $, if for all $x \ne a, x \in O$, as $x \to a$ the values $f(x)$ approach $L$, ($f(x) \to L$).
The limit, $L$, if such a number exists, is interpreted traditionally as...
  1. The number that is the best estimate for values of $f(x)$ when $x$ is a number close to but not equal to $a$;
  2. The position that an object approaches at times close to $a$, when moving on a line with its position at time $x \ne a$ determined by $f(x)$;
  3. On the graph of the function $f$, the second coordinate of a point that would fill in a hole in the graph of $f$ when considered omitting any actual point with first coordinate $a$.
The formal definition for $L$ to be called the limit says: Given any $\epsilon > 0$, there is a number $\delta >0$ with the property that if $a-\delta< x<a+\delta$ with $x \ne a$, then $L-\epsilon < f(x)< L + \epsilon$.
Using absolute values this condition is expressed:
Given any $\epsilon > 0$, there is a number $\delta >0$ with the property that if $0< |x-a|<\delta$, then $ |f(x)-L|< \epsilon$.
Visualizing the definition of limit number with mapping diagrams provides visual support to understand the information contained in this number for the function values for $x$ near $a$, i.e., $x \approx a$.

To illustrate the operation of the definition with a mapping diagram, consider an example function $f(x)=2x-\frac12$ with $a=1$, $L=1.5$ and $\epsilon = 0.5$.
Since in the definition the values $f(x)$ will be compared with $L$, we begin by visualizing $L$ on the target axis of the diagram, as well as the interval  $(L-\epsilon , L + \epsilon)$.
 
The choice of $\delta$, here we'll use $\delta =0.25$, is the essential feature of the definition, being made after the selection of $\epsilon$ and controlling the numbers $x$ to which the function is to be applied.
So the interval determined by $\delta=0.25$ is visualized on the domain axis of the diagram with the interval $(a-\delta, a+\delta)$.
After the choice of $\delta$ the task is to show that any $x \ne a$ in the interval $(a-\delta, a+\delta)$, must have $f(x) \in (L-\epsilon , L + \epsilon)$. Using the arrows to visualize a sampling of relevant $x$ and related $f(x)$ values gives a sense of what must hold true for $\epsilon = 0.5$.
 

But this quality must work not just for the chosen $\epsilon$, but this ability to find an appropriate value for $\delta >0$ must be possible for any $\epsilon>0$.
A dynamic visualization of the definition is found in CCD.DLCV, Dynamic visualization of the definitions of limit and continuity.