Formal Limit Definition

This is an illustration of the formal concept of a limit. One formal definition for a limit is: Let

be defined on an open interval about

, except possibly at

itself. We say that the limit of

approaches

is the number

, and write

, if, for every number

, there exist a corresponding number

such the for all

. If setting any goal

value you can make

small enough so that the above inequality holds the limit exist. There are two indications that

is small enough. The brown box shows

. The blue dash dot line show the minimum and maximum values of

within the

bounds. If both blue dash-dot lines cross the box the statement is true for the given

. Also, the values are shown with a nice blue equation. The red equation with yellow background indicates that

is not small enough.

For each function explore placing

at different values and decreasing

with the close button. Try placing

on boundary circles. Note:

can snap to grid values if the cursor is on the point.

For the first two functions ( floor and ceil ) does the limit exist at any integer value?
For all of the functions is there anywhere where the limit does not exist?
When is the point is not included in the Ymax and Ymin values?

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