limits and infinitesimals...
not really infinitesimal, but to us it may as well be. looks just like zero to us (according
to our instruments) but when we add it up a bunch of times, it is not zero. no matter how
accurrate our instruments are, it is always beyond their range of precision.
.99999999999999999999999999 = 1, right?
wrong... takes a while to figure out... keep squaring, eventually get a number that is
less than 1. but how many times do you square it? i just tack on more 9's, and you can't
tell the difference. so there is an infinitesimal difference. the number is not quite 1, but
it is 1 for all practical purposes. i can get it as close to one as i'd like. i can get it
out of YOUR range of detection. your instruments can't tell the difference, yet the number
acts differently than 1. Thus, it must be infinitesimally close to 1. rigorous? no. intuitive?
sorta :).
heart of epsilon-delta proofs
can get as close to the true value of F(x) as you want, without actually being f(x)
