This is an illustration of the formal concept of a limit. One formal definition for a limit is:[br][br]Let [math]f(x)[/math] be defined on an open interval about [math]x_0[/math], except possibly at [math]x_0[/math] itself. We say that the limit of [math]f(x)[/math] as [math]x[/math] approaches [math]x_0[/math] is the number [math]L[/math], and write[br][br][math]\lim_{x \to x_0} f(x) = L[/math] ,[br][br]if, for every number [math]\epsilon \gt 0[/math], there exist a corresponding number [math]\delta \gt 0[/math] such the for all [math]x[/math],[br][br][math]0 \lt | x - x_0 | \lt \delta \implies | f(x) - L | \lt \epsilon [/math].[br][br]If setting any goal [math]\epsilon[/math] value you can make [math]\delta[/math] small enough so that the above inequality holds the limit exist. There are two indications that [math]\delta[/math] is small enough. The brown box shows [math] x_0 \pm \delta \text{ and } L \pm \epsilon[/math]. The blue dash dot line show the minimum and maximum values of [math]f(x)[/math] within the [math]\delta[/math] bounds. If both blue dash-dot lines cross the box the statement is true for the given [math]\epsilon[/math]. Also, the values are shown with a nice blue equation. The red equation with yellow background indicates that [math]\delta[/math] is not small enough.

For each function explore placing [math]x_0[/math] at different values and decreasing [math]\delta[/math] with the close button. Try placing [math]x_0[/math] on boundary circles. Note: [math]x_0[/math] can snap to grid values if the cursor is on the point.[br][list][br][*]For the first two functions ( floor and ceil ) does the limit exist at any integer value?[br][*]For all of the functions is there anywhere where the limit does not exist?[br][*]When is the [math]x_0[/math] point is not included in the Ymax and Ymin values?[br][/list]