Here is a way to help your students understand deltas and epsilons. As I've said before I'm not that sure that being able to find the d-e relationship alone does that. So lets assume students are able to compute d-e in general and look at the lim(x-->3) (2x-5) = 1 where the usual approach will give d = e/2
Begin by drawing the "box," that is the rectangle whose vertical sides are x = a - d and x = a + d and whose horizontal sides are y = L+ e and y = L - e. Use a calculator or better yet a good graphing program like Winplot (or Geogebra, Nspire, etc.) Use sliders for the actual values of d and e.
CASE 1: With d = e/2 the box will be twice as high as it is wide. The graph of y = 2x - 5 will go exactly through the top left and lower right corners of the box. For all values of x between 2 - d and 2 + d the corresponding function value lie between 1 + e and 1- e. The requirements of the definition are met and we have proven that the limit is 1.
CASE 2: Let d = e/4. The box is now 4 times as high as it is wide. The graph enters and leaves through the sides of the box. For all values of x between 2 - d and 2 + d the corresponding function value lie between 1 + e and 1- e. The requirements of the definition are met and we have proven that the limit is 1.
CASE 3: Let d = e, the box is now a square. The graph enters the bottom of the box and leaves through the top. It is NOT true that for all values of x between 2 - d and 2 + d the corresponding function value lie between 1 + e and 1- e. There are some points outside the box top and bottom that should be between the horizontal lines but are not. The requirements of the definition are NOT met and we have NOT proven that the limit is 1.
It makes no difference what value of e is used. And this is important too. The fact that the box looks the same (technically a similar figure) for a particular delta is the graphical way of saying "for any e" ("for all e's", "for every e"). Large e, small e, the d = e/2 box is always twice as high has it is wide and the graph goes through the corners. This makes it easy to believe that as you go smaller and smaller the picture will not change ever.
With the sliders you can play with different d-e relationships for this or any function even if you cannot find algebraically the largest such relationship.
For polynomial and other functions where the relationship is something like d = min(1, e/3) the CASE 1 graph is not possible. The function may enter or leave at one corner but not the opposite corner; it will leave through the opposite vertical side meeting the requirements; it's more like Case 2. But with small values of d an e local linearity takes over and things look like Case 1. Taking e large enough so that d = 1 is instructive as well. As long as the graph enters and leaves through the sides of the box everything is wonderful.
Of course the box is the same box that we've all drawn on the board and is in all the textbooks. But the dynamic role played by the sliders is far more instructive than the dynamic hand waving with a static picture.
Hope this helps
