This applet demonstrates the precise 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.

This applet 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.
GeoGebra note: the highlighting doesn't always work; occasionally it will produce red and green highlighting simultaneously. (This is work from Dan Drake, modified to include LaTeX by Edward Kim.)