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.
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