The epsilon-delta definition of limit
This applet demonstrates the epsilon-delta definition of limit. Portions of the graph of f for which |f(x) - L| is *not* less than epsilon are highlighted in red. When the value of delta is "good enough" (that is, it satisfies the definition: if |x -a| is less than delta, then for such x, |f(x) - L| is less than epsilon) the relevant portion of the graph is highlighted in green. Different functions can be used by typing "f(x) = ...." into the input box.
I've never been very satisfied with the epsilon-delta applets I've seen. This applet is almost precisely what I've always wanted, since it highlights the values of f(x) that make your delta not work, and makes it clear when it does work. Mathematical note: the highlighting doesn't exclude x=a, so if you manage to give GeoGebra a function with a removable singularity at x=a, the highlighting won't turn green for any value of delta. However, this problem might be a pedagogical opportunity to reinforce the idea that the value of f(a) is irrelevant to the limit. GeoGebra note: the highlighting doesn't always work; occasionally it will produce red and green highlighting simultaneously. This usually happens when delta is *almost* small enough, and doesn't seem to be much of a problem.