A number $L$ is called

The limit, $L$, if such a number exists, is interpreted traditionally as...

- The number that is the best estimate for values of $f(x)$ when $x$ is a number
**close to and greater than** - The position that an object approaches at times close to $a$, when moving on a line with its position at time $x > a$ determined by $f(x)$;
- 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 points on the graph with first coordinate less than $a$.

This condition is also 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$.

A similar definition can be given for a number $L$ to be called

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 of $\lim_{x \to a^-} f(x) = L $ 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 $\delta=0.25$ for the left side limit is visualized on the domain axis of the diagram with the interval $(a-\delta, a)$.

So the interval determined by $\delta=0.25$ for the left side limit is visualized on the domain axis of the diagram with the interval $(a-\delta, a)$.

After the choice of $\delta$ the task is to show
that any $x$ in the interval $(a-\delta, a)$, 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 of $\lim_{x \to a^+} f(x) = L $ is found in CCD.DLCV, Dynamic visualization of the definitions of limit and continuity.

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 of $\lim_{x \to a^+} f(x) = L $ is found in CCD.DLCV, Dynamic visualization of the definitions of limit and continuity.