I have been following the current discussion of the difficulties of teaching limits with considerable interest.
Deltas and epsilons and the foundations of differential calculus do not attract much interest in the beginning calculus sequence these days. Yet in the beginning of this century there was great excitement (in scientific circles anyway...) over the extraordinary finesse proposed by Cauchy, Weierstrass, Dedekind and Co. to resolve some ancient and deep philosophical problems.
For example, Bertrand Russell was so astonished that he pronounced "...all goes smoothly until we reach those studies in which the notion of infinity is employed - the infinitesimal calculus and the whole of higher mathematics. The solution of the difficulties which formerly surrounded the mathematically infinite is probably the greatest achievement of which our age has to boast."
Nowadays, using quantifiers, the basic definitions of differential calculus can be expressed with (laudable) logical efficiency, yet this masks the excitement of discovering the triumph of human intellect celebrated by Russell.
I enclose excerpts below from a handout I use in my college (multisemester calculus) classes that attempts to recapture this experience; it has proved to be highly effective in class and makes the whole discussion of limits into an entertaining adventure. It may be useful to you all as well, for the high-school students I occasionally get in my classes are often the best in the class....
I would be happy to send the entire essay; send a note to my e-mail at
Howard Swann San Jose State University California
* * * * * * The Theory of Differential Calculus
Calculus provides a basis for much scientific reasoning. Mathematicians sought methods of proof for the validity of the procedures of calculus that are as indisputably correct as the demonstrations of Euclidean geometry, where, for example, each step of reasoning about congruence of triangles has a justification. Yet calculus tries to deal with concepts that are, in part, incomprehensible to the human mind. Our concept of number is curiously inadequate. We sense that there should be a number that represents the length of any line segment. The Greek Pythagoreans (c. 550 B.C.) discovered that the hypotenuse of a right triangle with legs of length 1 would necessarily have a length whose square was 2, yet there was no rational number, a number represented by the ratio of two integers, whose square was 2. This discovery was viewed with great alarm, for the Pythagoreans, a religious society, based their understanding of creation on the supposition that the integers and rationals described all that was `good', yet here was a (perfectly good) length that could not be described in such terms. According to legend, one of the Pythagoreans revealed this discovery to the Pythagorean's enemies, and was shipwrecked at sea as punishment. For the rest of us, this demonstration showed that some new `irrational' numbers representing such lengths were needed to extend the set of rational numbers. Straightforward algorithms for adding, multiplying and dividing rational numbers - fractions - are easy to justify; if we are to invent new irrational numbers we must find unquestionably correct ways to add, multiply and divide such new numbers.
Some 2400 years later in the 19th century, axioms were proposed that provided the structure to represent any length by a possibly infinite (non-terminating) decimal. The axioms also give compellingly plausible ways to define addition, multiplication and so forth for such infinite decimals. The set of infinite decimals together with the methods for doing the arithmetic operations is optimistically called the real numbers, although infinite decimals seem somewhat unreal, requiring that we become comfortable with a "number" representing the notion of adding an infinite number of rationals which, interpreted literally, cannot be done. The fundamentally incomprehensible concept of infinity keeps keeps coming up in one way or another.
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Our sense of a correspondence between numbers and locations along the `number line' leads to the idea that a certain trichotomy must be true; given any two numbers a and b, either they are equal, or a < b or b < a. The real numbers have this property; one just compares the digits in two "infinite decimals" to make the required determination. But the trichotomy property leads to confusion. Given any real number a, there can be no real number b next to a for the following reason: if there were such a b not equal to a, then a < b or b < a. In either case the new average number (a+b)/2 is closer to a than b. Thus the real numbers do not seem to model the `continuum'that we sense represents locations in space or the flow of time; there still seem to be points missing. If we insist that the real numbers possess the trichotomy property (and have the conventional order) we are stuck with this problem.
In view of these concerns, the extension of the rational numbers to the real numbers does not provide a completely satisfactory structure for a mathematical model for motion. We represent the motion of a point along a straight line by a function p(t), where t is a real number representing time, with real number p(t) giving the location on the number line at time t. But how is motion possible? If a point is located precisely at real number p(t) at time represented by real number t, by the argument above, there is no next instant later in time, nor is there any possible location next to p(t). Does the point momentarily disappear on its way to a later location p(t') at later time t' > t? If so, it disappears infinitely often.
In spite of this puzzle, motion clearly does occur somehow, and we go ahead and use a real valued function p(t) to represent motion, only to encounter a similar problem when we try to define what we mean by the instantaneous velocity of point p(t), which is the derivative of function p(t). We turn to the equivalent problem of defining what should be the slope of a line tangent to the graph of a function.
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Suppose we seek the derivative of y = 2^x at x = 0. Thus we seek a limit for T(h) := [2^h - 1]/h as h --> 0. We tabulate some values for T(h):
It appears from the table that T(h) is approaching some number close to .693???.... as h approaches zero through negative or positive values, but we don't really know what that limiting number might be.
We can expand the table above of values for T(h), letting h take smaller and smaller values and thus generate more and more digits of the apparent limit, but eventually our calculators will no longer be able to make accurate computations, or will produce numbers that no longer agree with the apparent trend of the computations. The following values were obtained from a CASIO fx-7500G: h 10E-5 10E-6 10E-7 10E-8 10E-9 10E-10 10E-11 T(h) .6931496 .693147 .69315 .6931 .693 .69 .00
We are uneasy accepting this termination of our efforts, for our answers should be independent of human notions of "small" (or large.) We may consider .0001 cm. to be "small", yet in some culture on some planet in some other galaxy it may be very "large". We have encountered the classical problem of dealing with the "infinitesimally small", a notion that defies attempts to give it a comfortable definition.
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From the examples above, graphically, the task of finding a limit for a function T(h) as h approaches 0 (assuming that there is a limit in some satisfactory sense) is the same as finding a number L so that, if you place a dot at point (0,L), the parts of the graph of T(h) for h < 0 and h > 0 are then joined up so that the resulting graph is smooth or `continuous' across the vertical line through (0,0).
The problem with giving this pictorial interpretation a formal definition is that our model of the real numbers does not allow us to convert the intuitive ideas to clearly defined terms. There are NO numbers on either side of 0; there is no way to "link" the two parts of the graph of T(h) together with some point (0,L); our model for the real numbers and the points on the graph of T(h) is "all full of holes."
We have encountered the fundamental conceptual problem with the real numbers once again. We do not seem to be able to reconcile two intuitively "obvious" properties that should be aspects of our sense of how numbers describe the continuum we call the number line. We cannot simultaneously retain a notion of smooth `continuous' one-dimensional motion of a particle and hold that the real numbers describing the location of the particle are `points', where we can determine if one point is less than another. The problem with the `holes' cannot be avoided. What is needed is a way to proceed that avoids these difficulties yet still is sure to get the right limiting numbers. The following finesse was proposed in the 19th century:
We consider the more general problem of determining whether or not there is a limit for some function g(x) as x approaches some number a. (a) What exactly would such a limiting `number' be? Is it a `real number'? If so, what is its value? (b) Is there some way of being sure that you have the right number?
Mathematician Karl Weierstrass (1815-1897) contrived the following way around the difficulties discussed above. Essentially, he shifted the burden back onto the function by proposing that we only discuss limits at points a for functions g(x) where the function g(x) must (1) provide a guess, a `real number' L as a limiting number for g(x) as x goes to a AND
(2) g(x) must provide an additional function that is guaranteed to have certain properties. (....delta(epsilon) will (eventually) be the name of this additional `function'...H.S.)
He was able to show that (i) these requirements are reasonable, and lots of functions satisfy these requirements and (ii) in spite of any confusion, the guess for the limit number for any function satisfying these requirements must be correct.
His arguments successfully avoided any reference to infinity or "infinitesimals" and, as we shall see, the reasoning is also unquestionably independent of any human perception of what was "small" or "close."
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The foundations of calculus and the many branches of mathematics that spring from calculus are rooted in this finesse of Weierstrass. In the proof of the (essential uniqueness) theorem (for limits) you will see how we manage to turn the profound difficulties concerning our concept of number to our advantage; as long as we stay within the class of functions defined to have limits in the sense of Weierstrass, the conclusions we make concerning the accuracy of our limits must be correct, even though we ultimately do not understand ...