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We are used to do limits by means of $\epsilon, \delta$. This is the first step an student encounters limits. Using $\epsilon$ and $\delta$ leads him to be familiar to Logic. I have taught limits and other basic similar concepts and I did that by using Maple instead of Mathematica. Because of that, writing a program in which we can probe a real function like sin$(x)$ has a limit has been my old wish. This needs a computer-assisted analytic approach in which we are able to define $\epsilon, \delta$ randomly in $\mathbb R$ and then verify if a certain well-behaved function has a limit. Here, I am not intended to use packages. Thanks.

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Are you asking for a visual look at the behavior? Or a computer-assisted analytic proof of something (maybe the Intermediate Value Theorem?) – Daniel Lichtblau Jun 19 '12 at 18:13
Welcome to Mathematica.SE. You will need to edit your question to provide a more solid description of your problem or it will be closed as "not a real question." If the community votes to close it you can still improve it and it may be reopened. Also, you have enough "reputation" to talk in Chat -- that may be helpful! – Mr.Wizard Jun 19 '12 at 18:33
@DanielLichtblau: Thanks for the comment and for your time. Honestly, I should have edited my question in time but I couldn't. As you kindly noted, I'd like to have a computer-assisted analytic approach rather than just seeing what happenes. In fact, how can we define $\epsilon, \delta$ in which a program be able to probe a certain limit? – Babak S. Jun 19 '12 at 19:56
If the edit to my answer goes in the direction you want, maybe you can add to your question some more details on what you mean by a "computer-assisted approach" - my interpretation of "enlivening" a theorem is not the same as "proving" a theorem. But one has to know more about how you want to illuminate the theorem. – Jens Jun 19 '12 at 20:03
up vote 2 down vote accepted

For visualizations, maybe the following Wolfram Demonstrations will give some genreal idea of what has been done:


For an analytical approach to derivatives using limits, you could just apply the technique I proposed in this more specialized answer:

It just uses the command Limit, as in this simple example:

Limit[(Sin[x + d] - Sin[x])/d, d -> 0]

(* ==> Cos[x] *)
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