What Is Limit?

The most basic concept of modern Calculus, that of limit, was never invoked by I. Newton and G. W. Leibniz, the creators of Calculus, even though it was implicit already in the works of Eudoxus and Archimedes. (It is interesting to follow the evolution of the concept of limit.)

Jean le Rond d'Alembert (1717-1783) was the first to propose [Dunham, p. 72] that Calculus is best based on the concept of limit, not controversial infinitesimals. But the first reasonably formal definition and consistent employment are due to Augustin-Louis Cauchy (1789-1857):

When the value successively attributed to a variable approach indefinitely to a fixed value, in a manner so as to end by differing from it by as little as one wishes, this last is called the limit of all the others.

Karl Weierstrass (1815-1897) gave the definition its (for a while) final $\epsilon -\delta$ form

$\displaystyle\lim _{x\rightarrow a}f(x) = L$ if and only if, for every $\epsilon \gt 0,$ there exists a $\delta \gt 0$ so that, if $0 \lt |x - a| \lt \delta ,$ then $|f(x) - L| \lt \epsilon $

which freed the notion of limit from its dependency on the idea of motion, replacing it with the concept of "nearness". (However, the expressions like "tends to" or "grows without bound" reflecting on change and motion are still very much in use even today.)

Weierstrass' definition works in any metric space endowed with a metric, say, $d:$

$\displaystyle\lim _{x\rightarrow a}f(x) = L$ if and only if, for every $\epsilon \gt 0,$ there exists a $\delta \gt 0$ so that, if $0 \lt d(x, a) \lt \delta ,$ then $d(f(x), L) \lt \epsilon .$

As a natural extension, function $f$ may be defined between two different metric spaces each equipped with its own metric. However, the idea of limit is rather topological than metric. Indeed, expressions like $|x - a| \lt \delta $ or $|f(x) - L| \lt \epsilon $ define neighborhoods of points $a$ and $L$ respectively, so that Weierstrass' definition could be read as

$\displaystyle\lim _{x\rightarrow a}f(x) = L$ if and only if, for every neighborhood $U$ of $L$ there exists a neighborhood $V$ of $a$ such that the images $f(x)$ of points $x\in V$ belong to $U,$ i.e. $f(V) \subset U.$

This definition makes sense (and is commonly used) for functions between two topological spaces. It easily incorporates the concepts of limit of function, as above, with a limit of sequence, provided we accept the idea of a neighborhood of "infinity" as a set of integers $n$ satisfying $n > N,$ for some $N.$ Since a sequence $\{a_{n}\}$ can be looked at as a function from the set $\mathbb{N}$ of natural numbers, we naturally get the formulation

$\displaystyle\lim _{x\rightarrow \infty }a_{n} = L$ if and only if, for every neighborhood $U$ of $L$ there exists a neighborhood of infinity (symbolically, that of $\infty)$ such that $a_{n}$ belongs to $U$ for all $n\in V,$ i.e., for all $n$ sufficiently large.

If $\{a_{n}\}$ is a sequence of real numbers, this admits a more common form

$\displaystyle\lim _{x\rightarrow \infty }a_{n} = L$ if and only if, for every $\epsilon \gt 0,$ there exists $N$ such that, for all $n \gt N,$ $0 \lt |a_{n} - L| \lt \epsilon .$

A sequence that has a limit is said to be convergent, or, more accurately, to be convergent to its limit. A sequence that does not have a limit is said to diverge or be divergent.

For mathematicians, I am sure, the $\epsilon -\delta$ definition, brief yet uniquely unambiguous, is a manifestation of mathematical beauty. For an average Liberal Arts student and their teachers, the definition is a stumbling block to be rather avoided. This is really a staggering problem: given $f,$ $a,$ and $\epsilon$ (or even $L),$ how do you find (or prove inexistence of) the corresponding $\delta ?$ Curiously, most mathematicians do not see this as a problem at all; not necessarily because a professional can resolve it effortlessly, but rather because the mere (knowledge of) existence of such perfect definitions imbues mathematics with order and consistency which fascinates mathematicians in the first place.

From the definition, one first derives a few fundamental properties of limits:

$\begin{align} \lim (Af(x) \pm Bg(x)) &= A\lim f(x) \pm B\lim g(x),\\ \lim (f(x) \times g(x)) &= \lim f(x) \times\lim g(x),\\ \lim (f(x) / g(x)) &= \lim f(x) / \lim g(x), \end{align}$

where $A,$ $B$ are constants and $\lim g(x) \ne 0,$ for the last identity. These (and a few other) properties allow finding and testing limits incrementally (or inductively) combining the limits of simple functions into the limits of more complex ones.

For example, for the sum of a geometric series $1, q, q^{2}, \ldots$ we wish to prove that, for $|q| \lt 1,$

$\displaystyle\lim _{n\rightarrow \infty } \sum _{k \le n}q^{k} = \frac{1}{1 - q}.$

As we know that $\displaystyle\sum _{k \le n}q^{k} = \frac{1 - q^{n + 1}}{1 - q},$ we can write (omitting $n\rightarrow \infty$ ):

$\displaystyle\begin{align} \lim \frac{1 - q^{n + 1}}{1 - q} &= \frac{\lim(1 - q^{n + 1})}{\lim (1 - q)}\\ &= \frac{\lim (1) - \lim q^{n + 1}}{1 - q}\\ &= \frac{1 - \lim q^{n + 1}}{1 - q}. \end{align}$

And all we need here is to establish that $\displaystyle\lim _{n\rightarrow \infty }(q^{n + 1}) = 0,$ for $|q| \lt 1.$

This is rather straightforward. Pick an $\epsilon .$ We wish to find an $N$ such that, for $n \gt N,$ $|q^{n + 1}| = |q|^{n + 1} \lt \epsilon .$ We shall use the fact that $\log (x)$ is a monotone increasing function: since $|q|^{n + 1} \lt \epsilon ,$ then (applying $\log$ to both sides of the inequality) $\log (|q|^{n + 1}) \lt \log (\epsilon ).$ But $\log (|q|^{n + 1}) = (n + 1) \log (|q|).$ This leads to $(n + 1) \gt \log (\epsilon ) / \log |q|.$ (We had to change "$\lt$" to "$\gt$" because, by our assumption $|q| \lt 1,$ so that $\log |q|$ is negative. Thus any $N$ that satisfies

$N \gt \log (\epsilon ) / \log |q| - 1$

will serve our purpose.

The simplest example of direct determination of $\delta ,$ given an $\epsilon ,$ is supplied by the linear function $f(x) = Ax + B.$ Here, one can take $\delta = \epsilon / A,$ but note that any smaller $\delta$ would do as well. Indeed, if $|x - a| \lt \epsilon / A$ then $|Ax - Aa| \lt \epsilon $ and hence $|(Ax + b) - (Aa + b)| \lt \epsilon .$ In other words, taking $L = f(a),$

$|f(x) - L| \lt \epsilon ,$

provided $|x - a| \lt \delta .$

By the way, the function $f$ that satisfies $\displaystyle\lim _{x\rightarrow a}f(x) = f(a)$ is said to be continuous at $a.$ The properties of limits tell us that the continuity of functions propagates via linear combinations, products and division (with some restrictions.) What we just proved is the continuity of the linear function. From this the continuity of rational functions, which are the ratios of polynomials, is automatic at all points where the denominator does not vanish. Thus the simply proved properties of the limits obviate the need of searching for a $\delta .$

Since $\displaystyle\lim _{x\rightarrow a}x = a,$ for continuous functions, the symbols of function and limit commute:

$\displaystyle\lim _{x\rightarrow a}f(x) = f(\lim _{x\rightarrow a}x).$

Similar interchange may sometimes be valid in more involved circumstances. For example, it is meaningful (even in more than one sense) to consider limit of a sequence of functions or curves. In this context, it is also meaningful to inquire which attributes of functions or curves commute with the symbol of limit. For example, if a sequence of curves $c_{n}$ has (in some sense) the limit $c,$ which is another curve: $\lim _{n\rightarrow \infty }c_{n} = c,$ and if each of the curves has a well defined length $len(c_{n}),$ is that true that the length of the limit curve $c$ can be found as the limit of the sequence of lengths:

$\displaystyle len (c) = len(\lim _{n\rightarrow \infty }c_{n}) = \lim _{n\rightarrow \infty }len(c_{n})$?

The answer is, Not necessarily! The limits must be approached judiciously. A frequent example where the limit of lengths does converge to the length of the limit curve, is the approximation of the circumference of a circle by the lengths of inscribed (or circumscribed, or both) broken lines, i.e., polygons. Such an approximation has been used from the ancient times for finding successive digits of $\pi .$ An apparently similar construction of approximations of a square by broken lines does not work. The limit of the lengths exists and the limit curve has the length but the two are different.

In a third example, each of the curves in a sequence has the length and the curves converge to a well-defined limit curve. The latter, however, can't be assigned length in a meaningful way. (Although it does possess a property expressed numerically by a different measure.)

It is worth noting that in all three examples a curve has been approximated by a sequences of broken lines.

References

  1. W. Dunham, The Calculus Gallery: Masterpieces from Newton to Lebesgue, Princeton University Press, 2008
  2. I. Grattan-Guinness, The Rainbow of Mathematics, W. W. Norton, 1997

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