What's the importance of "infinitesimally small" whenever calculus is explained First of all, i just like reading and understanding things related to math, but NOT at all any expert in math. So, I apologize if the question seems dumb.
It always puzzles me whenever a basic calculus is explained, it always make use of "infinitesimally" "very very small" etc. Why is it so important to use this term  in calculus? 
Similarly with "limits". The answer always seems blurry.. as it ends up with the notion of "the answer is say, as and as we go from point $a\to b$" etc. 
Why can't there be a "clear and perfect" answer when doing problems in calculus. Like as i said, why something unclear like "infinite" or symbols like "$x \to 2$" (limits) etc needs to be used in Calculus for solving the problems.
V.

How close do you want it? Pick any number greater than zero.
B: Ok, how about this number $\epsilon$ I happen to have in my pocket?