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, $L$, 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 but not e****qual****to** - 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 $f(x)$;
- 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 $a$.

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$.

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 $(a-\delta, a+\delta)$.

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.**
**

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.