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?
Best Answer
Here is a (possibly over simplified) way to look at continuity and differentiability for functions from $\mathbb{R}$ to $\mathbb{R}$.
Continuity means that there are no "breaks" or "jumps" in the graph of the function. If I were to draw the graph of the function, I would be able to do it without lifting my pen. Continuity also means that around every point, I can choose an interval small enough so that the function varies as little as I want.
A limit will be when we look at the behaviour around a point, but not necessarily at the point itself. It is very important to note that for a continuous function, $\lim_{x\rightarrow a} f(x)=f(a)$ for every point. (This can actually be an alternate definition for continuity)
Differentiability means that at every point, if we zoom in really really close, the function looks like a straight line segment. It means that the function will be continuous, and that there will be no sharp points in the graph of the function.
Hope that helps,
Remark: This answer may be a bit murky, and I personally suggest learning the $\epsilon$-$\delta$ to fully understand things for yourself.