Dear friends (and foes;-), limits are not needed to understand and do calculus. You can read my article at http://www.mathfoolery.com/Article/simpcalc-v1.pdf and my recent translation of a 1981 talk by V.A. Rokhlin at http://mathfoolery.wordpress.com/2011/01/01/a-lecture-about-teaching-mathematics-to-non-mathematicians/ I hope it will get you thinking.
The following is my response to fedja, it may be of interest to those who haven't read my article. First, let me indicate briefly my suggestions on how to approach calculus. The most important principle (of V.I. Arnold, explicitly stated by him in his recent Lectures on Partial Differential Equations) is to concentrate on examples, calculations, and applications, staying away from the generalities before they become necessary and the ideas behind them are well understood in concrete situations.
I will mostly discuss differentiation, see my article and my talk slides at http://www.mathfoolery.com/talk-2010.pdf for more details.
In high school they teach kids how to factor algebraic expressions, long and synthetic division of polynomials, the fact that if $f(a)=0$ then $x-a$ divides $f(x)$ (for a polynomial $f$). Why not use it and develop differentiation of polynomials? Indeed, if you ask a high school student to make sense of $\frac{x^2-a^2}{x-a}$ for $x=a$, (s)he is very likely just to factor the numerator, cancel $x-a$ and stick in $x=a$ to get $2a$ (or $2x$). This stuff is purely algebraic and all the differentiation rules are immediate.
Parenthetically, once they are established for polynomials, they are forced upon us for any reasonable understanding of differentiation, at least for uniformly differentiable because of the Weierstrass approximation theorem, for example. Also differentiation as an aspect of factoring becomes apparent.
We also can similarly develop differentiation of rational functions and roots, and use implicit differentiation to do other algebraic functions. This gives us already a lot to play with and to apply. The sine function can be treated geometrically, as an aspect of kinematics of the uniform rotation. This broadens the range of potential applications.
To get to the geometric and intuitive meaning of differentiation, we can notice that $x^n-a^n-na^{n-1}(x-a)$ has a double root at $x=a$, or we can look at the expression $f(x+h)$, $f$ being a polynomial, as a polynomial in $h$ with coefficients depending on $x$.
It has a constant term $f(x)$, the linear term $f'(x)h$ and all the higher order terms, so $f(x+h)-f(x)-f'(x)h=h^2r(x,h)$ where $r$ is a polynomial. This way, if we restrict $x$ and $h$ to some finite interval, we arrive at our basic estimate, uniform in $x$ and $h$:
$$|f(x+h)-f(x)-f'(x)h| \le Kh^2$$
that indicates how close is a polynomial to its affine approximation using its differential.
This inequality allows us to explain why polynomials with positive derivatives are increasing. We simply notice first that if $f' \gt C$ and $0 \lt h \lt C/K$, then $f(x) \lt f(x+h)$, and therefore $f(A) \lt f(B)$ when $A \lt B$. Then, by applying this fact to $f(x)+Cx$, we see that $f(B)-f(A) \gt C(A-B)$ for any $C \gt 0$ when $f' \ge 0$, and therefore $f(A) \le f(B)$. This is called the monotonicity principle, and it is the most complicated theorem in this approach to calculus. Everything else follows from it.
Now, to broaden our scope, we promote our basic estimate to the definition status and call the functions that satisfy this definition (uniformly) Lipschitz differentiable (LD). Derivatives of polynomials are polynomials, and differentiation of polynomials is related to their factoring. Likewise, derivatives of LD functions are Lipschitz. Indeed, we can rewrite our basic estimate as $|\frac{f(x)-f(a)}{x-a}-f'(a)|\le K|x-a|$, then notice that
$ \frac{f(x)-f(a)}{x-a} = \frac{f(a)-f(x)}{a-x}$ and conclude that
$|f'(x)-f'(a)| \le 2K|x-a|$. Moreover, $f$ is LD if and only if $f(x)-f(a)$ factors through $x-a$ in the class of Lipschitz functions of 2 variables, $x$ and $a$. Differentiation rules are straight forward.
I suggest to develop integration in parallel with differentiation (since they work and are understood better together) starting with simple examples of Newton-Leibniz theorem, and working our way to approximating definite integrals by approximating the integrands by the functions that are easy to integrate, say, piecewise-linear functions showing integrability of, say, Lipschitz function, positivity of integral and proving Newton-Leibniz.
Now I can get to fedja's objections. The Lipschitz theory takes care of all the piecewise-analytic functions, and that's almost everything that we deal with in elementary calculus. When we run into functions that don't fit into the Lipschitz theory ($x^{3/2}$, for example), we broaden our definitions by replacing $h^2$ in our basic estimate (with $|h|^{3/2}$ for our example). Then we observe that the theory still holds for the weaker estimates. The Holder estimates i.e. the ones we get by replacing $h^2$ with $|h|^{1+\gamma}$, $0 \lt \gamma \lt 1$ gives us much more room to play. All moduli of continuity are not needed for any problem involving only a finite number of functions, and it covers the vast majority of problems we encounter in calculus.
Now, in the classical treatment the extreme value theorem is used to prove the Lagrange theorem that is used to prove that a function with positive derivative is increasing. But in our approach we have a direct proof of this fact, so we don't need it. And we don't need the intermediate value theorem to prove the Newton-Leibniz since it can be proven directly using positivity of the integral. By the way, both of these theorems are non-constructive.
One may ask about minima and maxima within this approach. The monotonicity principle takes care of this topic, since it assures us that the point where the derivative changes its sign form plus to minus will be a local maximum; the similar obvious result is true for a local minimum.
Fedja also said that the inverse function theorem fails miserably within Lipschitz functions. My guess is that he was talking about the theorem that says that the inverse of a monotonic continuous function on a closed interval is continuous. This is true, of course, but it may be a good thing, since it raises our awareness of the fact that the inverses of very nice functions can be computationally horrendous. It also can be a motivation to consider some other moduli of continuity. As for the inverse function theorem about local invertibility of the differentiable functions, its treatment within the Lipschitz class is not much different from the standard, and is somewhat simpler.
In any case, fedja and I should probably take our dialogue elsewhere. Thanks for all the comments.
I also want to mention that similar approaches to calculus and introductory analysis have been tried with a good measure of success by Hermann Karcher at Bonn University and Mark Bridger of Notheastern University. See Karcher's lecture notes with an English summary at http://www.math.uni-bonn.de/people/karcher/MatheI_WS/ShellSkript.pdf and 2007 book "Real Analysis: a Constructive Approach" by Mark Bridger, where he defines differentiation via factoring of $f(x)-f(a)$ into $x-a$ in the class of continuous functions. Karcher said (in a recent e-mail to Dick Palais): "I taught my last Calculus course by first using only Lipschitz continuity. At the end of the first semester I reached uniform convergence of functions and continuity. From the second semester on the procedure was the standard one. The students liked it a lot, I still meet one or the other and they still smile." There is also a nice book by Peter Lax "Calculus with Applications," where he deals with uniform instead of the pointwise notions.
Added on 1/28/2011. I have just got an e-mail from Hermann Karcher. It says: "it was nice to hear from you again. I read Rokhlin's talk. Maybe one has to go even farther:
Sometimes I think, everybody needs his own explanation, and a successful teacher is good in guessing what each individual child needs.I also read your paper. You won't be surprised that I am familiar with your arguments. I am now retired for 7 years. My last three semester
beginners course was my most successful one. Today I attended the PhD colloquium of the student of a younger colleague. That student (and some of the younger people in the audience) had been in that last course of mine. They were happy to see me again and say how
much fun those three semesters had been. For various reasons I was then in a situation where I could completely ignore, how the standard course in analysis proceeds. During the first semester I did my own stuff,ending up at continuous functions and their uniform
convergence at the END of the first term. The next two semesters proceeded as usual - but all the fun we had came from building the foundations differently and the fun stayed with us. I wish you similar experiences. Don't try to convert too many grown ups, just enjoy teaching. Best regards --Hermann Karcher."
By the way, an English translation of Karcher's lecture notes is in progress. I also heard today via facebook form Ursula Weiss who is a math professor in Germany (we were both graduate students at Brandeis in the late 1980s) that she has just finished translating the first chapter.
