LIMITS BY CONTINUOUS APPROACH - FUNCTIONS AND LIMITS - What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

CHAPTER VI. FUNCTIONS AND LIMITS

§3. LIMITS BY CONTINUOUS APPROACH

1. Introduction. General Definition

In §2, Article 1 we succeeded in giving a precise formulation of the statement, “The sequence an (i.e. the function an = F(n) of the integral variable n) has the limit a as n tends to infinity.” We shall now give a corresponding definition of the statement, “The function u = f(x) of the continuous variable x has the limit α as x tends to the value x1.” In an intuitive form this concept of limit by continuous approach of the independent variable x was used in §1, Article 5 to test the continuity of the function f(x).

Again let us begin with a particular example. The function image is defined for all values of x other than x = 0, where the denominator vanishes. If we draw a graph of the function u = f(x) for values of x in the neighborhood of 0, it is evident that as x “approaches” 0 from either side the corresponding value of u = f(x) “approaches” the limit 1. In order to give a precise description of this fact, let us find an explicit formula for the difference between the value f(x) and the fixed number 1:

image

If we agree to consider only values of x near 0, but not the value x = 0 itself (for which f(x) is not even defined), we may divide both numerator and denominator of the expression on the right side of this equation by x, obtaining the simpler formula Clearly, we can make this difference as small as we please by confining x to a sufficiently small neighborhood of the value 0. Thus for image; and so on. More generally, if ε is any positive number, no matter how small, then the difference between f(x) and 1 will be smaller than ε, provided only that the distance of x from 0 is less than the number image. For if

image

then

| f(x) – 1| = | x2 |< ε.

f(x) – 1 =x2.

image

Fig. 168. u = (x + x3)/x.

The analogy with our definition of limit for a sequence is complete. On page 291 we made the definition, “The sequence an has the limit a as n tends to infinity if, corresponding to every positive number ε, no matter how small, there may be found an integer N (depending on ε) such that

| ana| < ε

for all n satisfying the inequality

nN.”

In the case of a function f(x) of a continuous variable x as x tends to a finite value x1, we merely replace the “sufficiently large” n given by N by the “sufficiently near” x1 given by a number δ, and arrive at the following definition of limit by continuous approach, first given by Cauchy around 1820: The function f (x) has the limit a as x tends to the value x1 if, corresponding to every positive numberε, no matter how small, there may be found a positive number δ (depending on ε) such that

|f(x)a | < ε

for all XX1 satisfying the inequality

| xx1| < δ.

When this is the case we write

f(x)a as xx1.

In the case of the function f(x) = (x + x3)/x we showed above that f(x) has the limit 1 as x tends to the value x1 = 0. In this case it was sufficient always to choose image.

2. Remarks on the Limit Concept

The (ε, δ)-definition of limit is the result of more than a hundred years of trial and error, and embodies in a few words the result of persistent effort to put this concept on a sound mathematical basis. Only by limiting processes can the fundamental notions of the calculus— derivative and integral—be defined. But a clear understanding and a precise definition of limits had long been blocked by an apparently insurmountable difficulty.

In their study of motion and change the mathematicians of the seventeenth and eighteenth centuries accepted as a matter of course the concept of a quantity x steadily changing and moving in a continuous flow toward a limiting value x1 · Associated with this primary flow of time or of a quantity x behaving like time they considered a secondary value u = f(x) that followed the motion of x. The problem was to attach a precise mathematical meaning to the idea that f(x) “tends to” or “approaches” a fixed value α as x moves toward x1.

But from the time of Zeno and his paradoxes the intuitive physical or metaphysical concept of continuous motion has eluded all attempts at an exact mathematical formulation. There is no difficulty in proceeding step by step through a discrete sequence of values a1, a2, a3 · · ·· · · But in dealing with a continuous variable x that ranges over a whole interval of the number axis it is impossible to say how x shall “approach” the fixed value x1 in such a way as to assume consecutively and in their order of magnitude all the values in the interval. For the points on a line form a dense set, and there is no “next” point after a given point has been reached. Certainly, the intuitive idea of a continuum has a psychological reality in the human mind. But it cannot be called upon to resolve a mathematical impossibility; there must remain a discrepancy between the intuitive idea and the mathematical language designed to describe the scientifically relevant features of our intuition in exact logical terms. Zeno’s paradoxes are a pointed indication of this discrepancy.

Cauchy’s achievement was to realize that, as far as the mathematical concepts are concerned, any reference to a prior intuitive idea of continuous motion may and even must be omitted. As happens so often, the path to scientific progress was opened by resigning an attempt in a metaphysical direction and instead operating solely with notions that in principle correspond to “observable” phenomena. If we analyze what we really mean by the words “continuous approach,” how we must proceed to verify it in a specific case, then we are forced to accept a definition such as Cauchy’s. This definition is static; it does not presuppose the intuitive idea of motion. On the contrary, only such a static definition makes possible a precise mathematical analysis of continuous motion in time, and disposes of Zeno’s paradoxes as far as mathematical science is concerned.

In the (ε, δ)-definition the independent variable does not move; it does not “tend to” or “approach” a limit x1 in any physical sense. These phrases and the symbol → still remain, and no mathematician need or should lose the suggestive intuitive feeling that they express. But when it comes to checking the existence of a limit in actual scientific procedure it is the (ε, δ)-definition that must be applied. Whether this definition corresponds satisfactorily with the intuitive “dynamic” notion of approach is a question of the same sort as whether the axioms of geometry provide a satisfactory description of the intuitive concept of space. Both formulations leave out something that is real to the intuition, but they provide an adequate mathematical framework for expressing our knowledge of these concepts.

As in the case of sequential limit, the key to Cauchy’s definition lies in the reversal of the “natural” order in which the variables are considered. First we fix our attention on a margin ε for the dependent variable, and then we seek to determine a suitable margin δ for the independent variable. The statement “f(x)a as xx1” is only a brief way of saying that this can be done for every positive number ε. In particular, no part of this statement, e.g. “xx1” has a meaning by itself.

One more point should be stressed. In letting x “tend to” x1 we may permit x to be greater than or less than x1, but we expressly exclude equality by requiring that xx1 : x tends to x1, but never actually assumes the value x1. Thus we can apply our definition to functions that are not defined for x = x1, but have definite limits as x tends to x1; e.g. the function image considered on page 303. Excluding x = x1 corresponds to the fact that, for limits of sequences an as n → ∞, e.g. an = 1/n, we never substitute n = ∞ in the formula.

However, as x tends to x1, f(x) may approach the limit a in such a way that there are values xx1 for which f(x) = a. For example, in considering the function f(x) = x/x as x tends to 0 we never allow x to equal 0, but f(x) = 1 for all x ≠ 0 and the limit a exists and is equal to 1 according to our definition.

3. The Limit of image

If x denotes the radian measure of an angle, then the expression image is defined for all x except x = 0, where it becomes the meaningless symbol 0/0. The reader with access to a table of trigonometric functions will be able to compute the value of image for small values of x. These tables are commonly given in terms of the degree measure of angles; we recall from §1, Article 2 that the degree measure x is related to the radian measure y by the relation image, to 5 places. From a four-place table we find that for an angle of

image

Although these figures are stated to be accurate only to four places, it would appear that

(1)   sin x/x → 1 as x → 0.

We shall now give a rigorous proof of this limiting relation.

From the unit circle definition of the trigonometric functions, if x is the radian measure of angle BOC, for image we have area of triangle image area of circular sector image (see p. 278) area of triangle image

image

Fig.169

Hence

sin x < x < tan x.

Dividing by sin x we obtain

image

or

(2)  image

Now image. Since sin x < x, this shows that

(3)  1– cos x< x2,

or

   1– x2 < cos x.

Together with (2), this yields the final inequality

(4) image

Although we have been assuming that image, this inequality is also true forimage, and (–x)2 = x2.

From (4) the limit relation (1) is an immediate consequence. For the difference between image and 1 is less than x2, and this can be made less than any number ε by choosing image.

Exercises: 1) From the inequality (3) deduce the limiting relation image

Find the limits as x→ 0 of the following functions:

image

4. Limits as x → ∞

If the variable x is sufficiently large, then the function f(x) = 1/x becomes arbitrarily small, or “tends to 0.” In fact, the behavior of this function as x increases is essentially the same as that of the sequence 1/n as n increases. We give the general definition: The function f(x) has the limit a, as x tends to infinity, written

f(x) → a as x → ∞,

if, corresponding to each positive number ε, no matter how small, there can be found a positive number K (depending on ε) such that

| f(x) – a | < ε

provided only that | x | > K. (Compare with the corresponding definition on p. 305.)

In the case of the function f(x) = 1/x, for which a = 0, it suffices to choose K = 1/ε, as the reader may at once verify.

Exercises: 1. Show that the foregoing definition of the statement

f(x) → a as x → ∞,

is equivalent to the statement

f(x) → a as 1/x → 0.

Prove that the following limit relations hold:

image

There is one difference between the case of a function f(x) and a sequence an. In the case of a sequence, n can tend to infinity only by increasing, but for a function we may allow x to become infinite either positively or negatively. If it is desired to restrict attention to the behavior of f(x)when x assumes large positive values only, we may replace the condition | x | > K by the condition x > K; for large negative values of x we use the condition x < — K. To symbolize these two methods of “one-sided” approach to infinity we write

x → +∞, x→ – ∞

respectively.