Abstract: We describe a different definition of "limit," which may be easier to understand than the definition given in most calculus books, though it is equivalent to that usual definition. Browser requirements: This web page uses the symbol font for some Greek letters and other mathematical symbols. Consequently, this web page will display correctly on most modern web browsers, but not all. For instance, it will probably display correctly on Windows 95 computers running Netscape 3, Explorer 2, or later browsers.

Funnels: A More Intuitive Definition of Limit

by Eric Schechter, Vanderbilt University

Our textbook, by Stewart, gives this imprecise definition: lim_x®x₀ f(x) = L is defined to mean that

we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to x₀, but not equal to x₀.

This imprecise definition has several possible interpretations, and so the student is left to guess which interpretation is the right one. The student is aided by the several examples that accompany the definition. For proofs, the imprecise definition is inadequate, and ultimately a ``precise definition'' must be given; Stewart does so 20 pages later. Evidently he waited that long because he wished to avoid intimidating the student; the precise definition is much more complicated. It states that lim_x®x₀ f(x) = L means

for each number e > 0 there exists a corresponding number d > 0 with the property that, whenever x is a number with 0 < |x–x₀| < d, then |f(x)–L| < e.

I will refer to this as the epsilon-delta definition. It is the classical definition of limit (for a real-valued function of a real variable). Of course, different limit assertions may require different choices of epsilons and deltas: We could use one system of epsilons and deltas for the assertion lim_x®3 (2x–5) = 1, and another system of epsilons and deltas for the assertion lim_x®0 (sin x)/x = 1.

Many (perhaps most) calculus students have difficulty understanding and learning the epsilon-delta definition of a limit. I can state several reasons why the epsilon-delta definition is difficult to understand (although the student doesn't need to be aware of these reasons): it has too many variables; it has too many nested clauses; it does not suggest anything about a rate of convergence; and it cannot be illustrated easily with a picture. I sometimes tell my students to memorize the epsilon-delta definition, word for word; understanding will come later (if at all). I caution the students to be careful with their memorizing; students who do not yet fully understand the definition may inadvertently change the wording slightly, in some fashion that sounds inconsequential to the untrained ear but greatly changes the mathematical content.

In the paragraphs below, we shall consider an alternate definition, which may be easier for students to understand; we will call it the funnel definition of limits. (This experimental approach is modified from some as-yet unpublished material by Professor Bogdan Baishanski of Ohio State University.) The epsilon-delta definition and the funnel definition are equivalent, as we shall demonstrate at the end of this document. The proof of equivalence requires some understanding of subsequences, but no other specific knowledge beyond calculus.

The funnel definition of ``limit'' is in two steps; first we must define a ``funnel.''

Definition. A funnel is an increasing function j, whose domain and range are of the form (0,a) and (0,b) respectively, where a and b are any members of (0,+¥].

Examples.

j(t) = t,	considered as a function from (0,+¥) to (0,+¥)
j(t) = t,	considered as a function from (0,2) to (0,2)
j(t) = t²,	considered as a function from (0,+¥) to (0,+¥)
j(t) = Öt,	considered as a function from (0,+¥) to (0,+¥)
j(t) = t/(1+t),	considered as a function from (0,+¥) to (0,1)
j(t) = tan(t),	considered as a function from (0,p/2) to (0,+¥)
j(t) = 1–cos(t),	considered as a function from (0,p/2) to (0,1)

It is easy to draw pictures of some funnels:

I have chosen the name "funnel" to emphasize that the graph of j(t) is narrow (i.e., j(t) is small) for t near 0, like a kitchen funnel turned on its side; this is perhaps best illustrated by the funnel j(t) = t². That property is not immediately evident in the graph of Öt, which appears to be flat at its left end, rather than pointed. Nevertheless, Öt is an increasing function, so Öt is indeed a funnel.

Actually, the functions t, t², Öt will suffice for most early applications. The beginning student does not need to be burdened with exercises such as "prove that ln(1+t) is a funnel." However, such proofs are not particularly difficult: 1+t is an increasing function, and ln is an increasing function, hence ln(1+t) is an increasing function. Moreover, it may be useful to mention at the outset that a wide variety of functions can be used as funnels, and that particular funnels can be devised for particular applications.

Observation. Suppose that j is a funnel, with domain (0,a) and range (0,b). Suppose that 0 < a₁ < a, and let b₁ = j(a₁). Then j, considered as a function from (0,a₁) to (0,b₁), is also a funnel. Summarizing this result a bit imprecisely, we say that any funnel, restricted to a smaller domain, is also a funnel. That fact is important for some applications. Moreover, in most applications, it doesn't matter what size domain (0,a) we use, as long as a > 0 and a is small enough to satisfy the particular application; generally any smaller positive number will also work. Consequently, in many applications, we don't bother to specify the domain.

Definition. The equation lim_x®x₀ f(x) = L (in the sense of funnels) means that there exists a funnel j with the property that,

whenever x is a real number such that |x–x₀| is in the domain of j, then x is in the domain of f and |f(x)–L| < j(|x–x₀|).

Of course, different limit assertions may require different funnels. For instance:

lim_x®3 (2x–5) = 1. Proof: Take j = 2t.
lim_x®0 (sin x)/x = 1. Proof: Take f(x) = (sin x)/x for x¹0, x₀=0, L=1, and j = 1–cos(t) for 0 < t < p/2. Using geometry, we can prove that cos(x) < x^-1sin(x) < 1 when 0 < |x| < p/2 (see Stewart, page 171; no one claimed that the proof would be obvious!). Hence 0 < 1 - x^-1sin(x) < 1 - cos(x) when 0 < |x| < p/2 Note that cos(x) is an even function of x. It follows that |f(x)–L| = 1– x^–1(sin x)} < j(|x–x₀|) when |x–x₀| is in the domain of j.

A drawback to the funnels definition. The funnels definition may be easier to understand, but it is more difficult to use in proving some theorems. For instance, a basic theorem about limits is the Addition Theorem: if lim_x®x₀ f(x) = L and lim_x®x₀ g(x) = M, then lim_x®x₀ [f(x)+g(x)] = L+M. This is easy to prove using the epsilon-delta definition of limit (once one has understood that definition), but difficult using the funnel definition. What one needs to show is that the sum of two funnels is a funnel. That's true, but not easy to prove. If j and y are funnels, then it is clear that j+y is increasing and has domain equal to the intersection of the domains of the two given funnels, but it is not obvious that the range of j+y is an interval. Perhaps it would be best to switch to the epsilon-delta definition of limit (via the proof given below), before trying to prove anything like the Addition Theorem.

Appendix: Equivalence of the two definitions. We shall show that the epsilon-delta definition and and the funnel definition are equivalent -- i.e., an assertion of the form lim_x®x₀ f(x) = L is true with one definition if and only if it is true with the other definition.

In one direction, the implication is fairly easy: Suppose we are given some funnel j : (0,a) ® (0,b) that proves lim_x®x₀ f(x) = L in the sense of funnels; we wish to prove lim_x®x₀ f(x) = L in the sense of epsilons and deltas. Given any e > 0, choose

$\begin{displaymath}\delta\quad =\quad\left\{\begin{array}{lll} \varphi^{-1}(\va... ...2b\right)&\mbox{\rm if}&\varepsilon\ge b .\end{array} \right.\end{displaymath}$

Then 0 < |x–x₀| < d Þ |f(x)–L| < e. (This proof would be more complicated if we used a more general definition of "funnel.")

The implication in the other direction is harder. Assume lim_x®x₀ f(x) = L in the sense of epsilons and deltas. We shall prove lim_x®x₀ f(x) = L in the sense of funnels -- i.e., we shall find a suitable funnel.

Let (e_n : n=1,2,3,...) be any sequence of positive numbers, strictly decreasing to 0. For each positive integer n, there is some number d_n > 0 such that

0 < |x-x₀| < d_n Þ |f(x)-L| < e_n.

We now consider two cases, according to whether the d_n's are bounded below by some positive constant d.

Case (i): There is some constant d > 0 with the property that d_n > d for all n. In this case, we have

0 < |x-x₀| < d Þ |f(x)-L| < e_n.

Since the sequence (e_n : n=1,2,3,...) decreases to 0, it follows that 0 < |x – x₀| < d Þ |f(x)–L| = 0. In this case, any funnel will do, if we restrict it to a domain no larger than (0,d).

Case (ii): There is no such constant d > 0. In this case, by replacing (d_n : n=1,2,3,...) with a suitable subsequence, we may assume that

d₁ > d₂ > d₃ > ×××

and that the sequence (d_n : n=1,2,3,...) converges to 0. Replace (e_n : n=1,2,3,...) with the corresponding subsequence. Now define j so that its graph over each interval [d_n, d_n–1] is a straight line segment, with values at the endpoints given by j(d_n) = e_n–1 for n = 2,3,4,... . It is easy to verify that this makes j : (0,d₂) ® (0,e₁) a funnel, and 0 < |x–x₀| < d₂ Þ |f(x)–L| < j(|x–x₀|).