> - Why are limits defined the way they are (with epsilons and deltas)?

> - The book will probably touch lightly upon the Mean Value Theorem -- why is this important? What's the point?

> - Why is the chain rule true? It reads dy/dx = (dy/du) (du/dx). Yay! This is just cancelling fractions, right? Any "respectable" calculus book will insist that it's not, but most students will cheerfully ignore this, still get correct answers to the homework problems, and sleep fine at night.

Which "respectable" book(s) would you recommend for those who want to dive into this details? Is Tom Apostol's Mathematical Analysis? good for learning these kind of details? (They say this book is "respectable", but I would like to hear your thoughts about it. Thanks).