How is the concept of the limit the foundation of calculus?

Question

My casual study of mathematics and calculus introduced me to the notion that calculus didn't find a firm foundation until Cauchy, Weierstrauss (et al) developed set theory some ~100 years after Newton and Leibniz.

The concept of limits (and their epsilon-delta proofs) was what allowed calculus to get past the shaky logic of infinitesimals.

(The above is just me laying out what I believe to be true, but please feel free to take issue with any/all of it.)

HERE IS MY ACTUAL QUESTION (more or less).

How exactly do limits save calculus?

When thinking about the derivative, we want to contemplate the rate of change of a function wrt its input. Instead of using an infinitesimal change in the input, we take the limit as the change in the input goes to zero.

When we say "take the limit" we mean finding a band around the function value (+/- "epsilon") and derive a band around the input value (+/- "delta") such that any value of the input beyond a certain point is guaranteed to generate a function value within our desired band (i.e., +/- epsilon). This is all rather loose, but hopefully you catch the drift.

BUT HOW DOES THIS HELP SAVE CALCULUS?

Is it because by saying we can take any ARBITRARY epsilon, we are in effect, saying we can take EVERY epsilon > 0? We exhaust every epsilon all the way down to (but not including) 0. This action reminds me a lot of the idea that 0.999... = 1.

Sorry if this isn't clear, but if I had to put this in simplest terms, I guess it boils down to: is "arbitrarily many" the same as "infinitely many"?

Answer 1

Firstly, Calculus was originally formulated with "heuristic" infinitesimals. This was why limits were created. To try and remedy the lack of rigor present in the original arguments with infinitesimals. Secondly, now Calculus has been formulated with rigorous infinitesimals, so the view point that modern infinitesimal calculus is not rigorous is incorrect.

I should at least mention that limits have some odd intuitive gaps in their formulation. What I mean is that, for instance, taking the limit of a function as it approaches infinity is a perfectly rigorous operation, however for the limit to exist, both the left and right side limits must be equal. $$\lim_{x \to \infty} f(x)=\lim_{x \to \infty^+} f(x)=\lim_{x \to \infty^-} f(x)$$ The thing I find odd to this day, with only limits at infinity, is that the notation of approaching infinity from different directions is acceptable. To me at least, this seems to be intuitive nonsense seeing as infinity is: a) not a number in the first place and b) it can't be approached from a greater real number seeing as it is larger than all reals, i.e. the numbers we use in real analysis.

In the end, limits are easier to formulate than infinitesimals. If you don't like the epsilon-delta definitions, you should see the look of horror on some peoples faces when you merely say the word "ultra filter", its enough to say that's how most infinitesimals and transfinite numbers (different sized "infinities") are constructed rigorously. Infinitesimals could be used without evoking horrific derivations and definitions, much like how we teach children about addition and multiplication without justifying it with rings. Sadly, most people have a poor understanding of infinitesimals (I'm no expert either), so they end up not understanding the technicalities of saying that $1 \not = 0.999..$. It should be noted that most Nonstandard Systems (that's what they are called) do accept equality while others have rigorous arguments saying they differ by an infinitesimal. It's semantic black hole when you really understand it...

The Point

Learn limits and the epsilon-delta way, even considering I find the system somewhat flawed. If you really find this kind of thing rubbing you the wrong way, you can investigate other ways of formulating calculus, and by that point you'll hopefully be educated enough to have developed intuition for what you want the system to be like.

Answer 2

Another way to avoid limits is to consider dual numbers, where there is a number $\epsilon \neq 0$ with a vanishing square $\epsilon^2 = 0$. This gives an algebraic model of calculus because one may derive Taylor's theorem and the derivatives of elementary functions by pushing the argument into the dual domain.

Answer 3

The reason the $\epsilon$-$\delta$ definition work is because it's arbitrarily precise. We can go as close as we want and no matter how close we go the relation can be established. The issue with infinitesimal in the old days was that it was ill-formed so it easily lead to contradictions, the limit definition of $\epsilon$-$\delta$ made it certain it was easy to work with it and not result in any form of internal contradiction.

Today however there are infinitesimals that are well-formed such that they work. There are ways to use them to reformulate calculus.

Answer 4

Note: The concept of a limit had an enormous unifying impact in the beginning of the $19$th century at a time when there was a dissatisfaction with the logical status of analysis.

Some reasons were according to Morris Klines Mathematical Thought from Ancient to Modern Times; chap 40: The Installation of Rigor in Analysis :

• The very concept of a function was not clear, the use of series with regard to convergence and divergence had produced paradoxes and disagreements; the controversy about the representations of functions by trigonometric series had introduced further confusion and the fundamental notions of derivative and integral had never been properly defined.

• He cites Abel who complained in a letter of $1826$ about:

  the tremendous obscurity which one unquestionably finds in analysis. It lacks so completely all plan and system that it is peculiar that so many men could have studied it. The worst of it is, it has never been treated stringently. There are very few theorems in advanced analysis which have been demonstrated in a logically tenable manner. Everywhere one finds this miserable way of concluding from the special to the general and it is extremely peculiar that such a procedure has led to so few to the so-called paradoxes.

Rigorous analysis begins with the work of Bolzano, Cauchy, Abel and Dirichlet and was furthered by Weierstrass. We concentrate on Cauchy who introduced the concept of a limit systematically into analysis.

Cauchy's program:

Central to Cauchy's successful rigorization of the calculus was his simultaneous realization of two facts. First, that the eighteenth-century limit concept could be understood in terms of inequalities (given an epsilon, to find an $n$ or a delta).

Second, and more important, that once this had been done, all of the calculus could be based on limits thereby transforming previous results on continuous functions, infinite series, derivatives, and integrals into theorems in his new rigorous analysis. Though there were occasional gaps in his reasoning, he nevertheless far outdistanced his predecessors. And his work provided the necessary groundwork for the eventual complete rigorization of analysis by the school of Weierstrass.

Cauchy's sources:

The techniques of the algebra of inequalities came in large part from the works on approximations, Lagrange's systematic Equations numériques. Most of the other information Cauchy needed, and used, came from four other works from Lagrange and S.F. Lacroix. Lacroix's procedure in treating a topic was to summarize all the major work on it by the leading mathematicians. He, like most mathematicians of the time, wanted to show how to solve problems, therefore his Traité included whatever techniques were applicable to this end. Precisely because Lacroix's book is a mathematical museum of diverse methods and results, presented in full complexity, it could be of service to Cauchy.

Cauchy's definition of limit:

Cauchy defined the limit concept in these words:

When the successivley attributed values of the same variable indefinitely approach a fixed value, so that finally they differ from it by as little as desired, the last is called the limit of all the others.

This concept, translated into the algebra of inequalities, was exactly what Cauchy needed for his calculus. The very language of this verbal definition is sometimes taken to show the superiority of Cauchy's limit concept over all previous work.

Cauchy's definition is

• free from the idea of motion

• it does not depend on geometry

• it does not retain the unnecessary restriction, often included in the earlier definitions, that a variable could never surpass its limit.

This and much more information can be found in The Origin of Cauchy's Rigorous Calculus by Judith V. Grabiner.

Epilogue: One of the many benefits of the new limit concept was, that formerly difficult tasks could then be solved in a nearly automated process. I'd like to show at least one example, which nicely demonstrates the power and elegance of limits.

Let's have a look at the interesting improper integral

\begin{align*} \int_{0}^{\infty}\frac{1}{x^3-1}=-\frac{\pi \sqrt{3}}{9} \end{align*} We observe, that the integrand has a singularity at $x=1$ and it is blowing-up to minus infinity if $x$ approaches $1$ from values less than $1$ and it is blowing-up to plus infinity if $x$ approaches $1$ from values greater than $1$. But, what about the nice finite value on the RHS? It seems, that since the integral is signed, the infinities cancel away somehow. But how to show this rigorously? That's where the limits come into play!

Note that partial fraction decomposition gives

$$\frac{1}{x^3-1}=\frac{1}{3(x-1)}-\frac{2x+1}{6(x^2+x+1)}-\frac{1}{2\left[\left(x+\frac{1}{2})^2+\frac{3}{4}\right)\right]}$$

And integrating the individual terms on the RHS gives

\begin{align*} \int\frac{dx}{x^3-1}&=\frac{1}{3}\ln(x-1)-\frac{1}{6}\ln(x^2+x+1)-\frac{1}{\sqrt{3}}\tan^{-1}\left(\frac{2x+1}{\sqrt{3}}\right)\\ &=\frac{1}{6}\ln\left[\frac{(x-1)^2}{x^2+x+1}\right]-\frac{1}{\sqrt{3}}\tan^{-1}\left(\frac{2x+1}{\sqrt{3}}\right)\\ \end{align*}

The argument of the $\log$ function in the integration interval is well-behaved except at $x=1$ where we get $\ln(0)$. Therefore we consider

\begin{align*} \int_{0}^{\infty}\frac{dx}{x^3-1}&=\lim_{\varepsilon\rightarrow 0}\int_{0}^{1-\varepsilon}\frac{dx}{x^3-1} +\lim_{\varepsilon\rightarrow 0}\int_{1+\varepsilon}^{\infty}\frac{dx}{x^3-1}\\ &=\lim_{\varepsilon\rightarrow 0}\left(\left.\left[\frac{1}{6}\ln\left[\frac{(x-1)^2}{x^2+x+1}\right]-\frac{1}{\sqrt{3}}\tan^{-1}\left(\frac{2x+1}{\sqrt{3}}\right)\right]\right|_0^{1-\varepsilon}\right)\\ &+\lim_{\varepsilon\rightarrow 0}\left(\left.\left[\frac{1}{6}\ln\left[\frac{(x-1)^2}{x^2+x+1}\right]-\frac{1}{\sqrt{3}}\tan^{-1}\left(\frac{2x+1}{\sqrt{3}}\right)\right]\right|_{1+\varepsilon}^{\infty}\right)\\ &=\lim_{\varepsilon\rightarrow 0}\left(\frac{1}{6}\ln\left[\frac{\varepsilon^2}{(1-\varepsilon)^2+(1-\varepsilon)+1}\right]\right.\\ &\qquad-\left.\frac{1}{\sqrt{3}}\tan^{-1}\left(\frac{2(1-\varepsilon)+1}{\sqrt{3}}\right) +\frac{1}{\sqrt{3}}\tan^{-1}\left(\frac{1}{\sqrt{3}}\right)\right)\\ &+\lim_{\varepsilon\rightarrow 0}\left(-\frac{1}{\sqrt{3}}\tan^{-1}(\infty)-\frac{1}{6}\ln\left[\frac{\varepsilon^2}{(1+\varepsilon)^2+(1+\varepsilon)+1}\right]\right.\\ &\qquad\left.+\frac{1}{\sqrt{3}}\tan^{-1}\left(\frac{2(1+\varepsilon)+1}{\sqrt{3}}\right)\right)\\ &=\lim_{\varepsilon\rightarrow 0}\left(\frac{1}{6}\ln\left[\frac{\varepsilon^2+3\varepsilon+3}{\varepsilon^2-3\varepsilon+3}\right]\right)-\frac{1}{\sqrt{3}}\tan^{-1}\left(\sqrt{3}\right)+\frac{1}{\sqrt{3}}\tan^{-1}\left(\frac{1}{\sqrt{3}}\right)\\ &-\frac{1}{\sqrt{3}}\tan^{-1}(\infty)+\frac{1}{\sqrt{3}}\tan^{-1}\left(\sqrt{3}\right)\\ &=\frac{1}{\sqrt{3}}\tan^{-1}\left(\frac{1}{\sqrt{3}}\right)-\frac{1}{\sqrt{3}}\tan^{-1}(\infty)\\ &=\frac{1}{\sqrt{3}}\left(\frac{\pi}{6}-\frac{\pi}{2}\right)\\ &=-\frac{\pi\sqrt{3}}{9} \end{align*}

I found this example in Paul J. Nahin's Inside Interesting Integrals

Answer 5

There are two aspects which I will emphasize here:

1) Chronological Development of Calculus: When Newton and Leibniz invented calculus, it was based on intuitive (for some it was counter-intuitive too) notion of infinitesimals. These were supposed to be quantities which were non-zero but had their magnitude less than any pre-assigned values. As can be seen the statement above is self-contradictory. I think it is best to explicitly show the problem with infinitesimals by an example.

Suppose we have to find the derivative of $x^{2}$ with respect to $x$. According to Newton this is the ratio $$\frac{(x + \delta x)^{2} - x^{2}}{\delta x}$$ where $\delta x$ is a single (i.e. it is not the product of $\delta $ and $x$) infinitesimal quantity. On simplification we will get the value of this ratio as $(2x + \delta x)$ and since $\delta x$ is infinitesimal, the result is $2x$. So the idea was that infinitesimals were only a tool and would never appear in the final solutions to actual problems. In other words derivatives and integrals were calculated using infinitesimals but they themselves were not infinitesimals. The fundamental problem in the example above is that there is no precise way to know when to treat $\delta x$ as non-zero and when to treat it like zero.

There was no rigorous theory of infinitesimals so to speak and guys like Newton/Leibniz knew how to avoid the common pitfalls in this development of calculus (in other words they knew when to treat infinitesimals as zero and when not to treat them as zero). Later many analysts like Cauchy/Weierstrass tried to work their way around the contradictions of infinitesimals and developed a theory of limits. It was then that processes of differentiation and integration became a sort of limit. Much later in 1960s Abraham Robinson gave a sound theory of infinitesimals which was called Non-standard Analysis.

2) Theory of Real Numbers: While many answers have emphasized my previous point, I don't see any discussion on the role of Real Numbers. It has to be understood very clearly that a sound theory of real numbers was developed after the notions of limit and $\epsilon,\delta$ definitions were given. Thanks to work of Cantor/Dedekind, there was a system of numbers in which one could develop interesting analysis. Without the real number system one can define limits via $\epsilon,\delta$ (for example in $\mathbb{Q}$) but it will not have any of the interesting features which we have in calculus/analysis. This is because many limits will turn out to be outside $\mathbb{Q}$. Later authors of analysis have so de-emphasized the theory of real numbers that it is now being taken as granted (the axiomatic approach to real numbers).

In my opinion the foundations of calculus are made not only from a theory of limits but also from a theory of real numbers.

Answer 6

The OP asked (1) how the limits save the calculus, and more specifically (2) how the epsilon-delta definitions save the calculus. The distinction is well-taken and is helpful in clarifying the issues.

It is well known that there are two main tracks for the development of the calculus: (A) the approach exploiting an Archimedean continuum, where for example limits and continuity are defined via epsilon-delta; and (B) the approach exploiting what could be called a Bernoullian continuum, i.e., a continuum featuring infinitesimals. Johann Bernoulli was the first one to use systematically infinitesimals and differentials to found the calculus, whereas his teacher Leibniz exploited both (A-track) exhaustion techniques and (B-track) infinitesimal techniques.

The distinction between the OP's questions (1) and (2) is significant because limits are actually present in both the A-track and the B-track. Thus, one can define the limit of a function $f(x)$ as $x\to 0$ via epsilon, delta, but one can also define it as the standard part of $f(\alpha)$ where $\alpha\not=0$ is an infinitesimal.

This is a point that is often overlooked in historical commentary as well as in some of the answers above. Thus, to Cauchy the primitive concept was that of a variable quantity, and he defined both limits and infinitesimals in terms of this primitive concept. Cauchy's definition of continuity of $f$ is not at all what one would expect it to be based on traditional historiography. Cauchy says that $f$ is continuous if every infinitesimal increment $\alpha$ always produces an infinitesimal change $f(x+\alpha)-f(x)$ in the function.

To answer the OP's question, the reason epsilon-delta definitions so to speak "save" the calculus is because they are formulated in an A-track fashion. This "saves" the calculus because mathematicians around 1870 have not yet developed a theory of infinitesimals that was thought satisfactory. Therefore some of them thought that the A-track was the only one available. As mentioned in some of the other answers, this foundational difficulty was overcome about 60 years ago by Abraham Robinson, so today it is the infinitesimal B-track that "saves" the calculus as far as both the students and the professors are concerned.