I had my first encounter with Calculus a decade ago. Back then it was purely mechanical. Formulas and rules of derivation and integration were being written on the board without deriving it and were told to compute a bunch of derivatives and integrals without even having a notion of functions, limits and other essential concepts. Today i have a burning desire to relearn all the forgotton math, calculus in particular.
I'm having a bit of concern with the notion of Limits and Continuity. Is it really essential that i have solid understanding of limits and continuity? I could teach myself differential calculus after analytically computing limits as well as L'Hopital's rules and up to partial derivatives and basic integrals up to trig substitution. Frankly speaking I still don't have a solid understanding of Limits.
Please explain what a limit is in layman's terms, preferbly with an easy to understand analogy. I looked in wikipedia and searched for a lot of youtube videos, but couldn't make sense. Also explain Continuity in simple terms.
I would also like to know a little bit about Linear and quadratic approximations.
Many people agree that it's indeed possible to keep differentiating and integrating functions without even knowing what a limit is! Big question is why is it considered as a central idea of calculus?