Limits

[lyr]

I've been having a lot of grief lately with limits and therefore the foundations of calculus. It's not that I don't understand what limits are. I could move on perfectly well not knowing why a limit works, just knowing the rules for evaluating them. My problem is that I don't understand the details of how and why they work. I understand the

Comments

[booksoverbombs]

So, here's how it goes down: by Newton's motivation of the difference quotient, the derivative of the function, say, x^2 is ((x+dx)^2-x^2)/dx, or 2x + dx. But so long as we evaluate the expression with dx != 0, the value of the derivative isn't 2x, it's 2x plus some junk. If we evaluate with dx = 0, we've committed the sin of dividing by zero, per the original expression. Berkeley raises this objection in The Analyst, perhaps one of the most pedantic essays ever written by an author with a good point in mind. It wasn't until Robinson that infinitesimals were restored to their original mathematical footing.

In the interim, Weierstrauss developed the limit, a mathematical structure which did the heavy lifting of the infinitesimal without rousing Berkeleyan criticism. His insight was to recognize the congruences between the intuitive understanding of infinitesimal quantites, and the structure of the epsilon-delta definition of the limit. All the existence of a limit asserts is that if the lim (x->c) f(x) = y, then if you specify some real number ε, I can give you a number δ such that all points from f(c-δ) to f(c+δ) (excepting possibly f(c) itself) fall within [f(c)-ε, f(c)+ε]. In other words, for any number, no matter how small, there's a neighborhood of points in the domain around c such every associated point in the range is as close to the limit as you've specified.