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 x_{0}, but not equal to x_{0}. |
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 ε > 0 there exists a corresponding number δ > 0 with the property that, whenever x is a number with 0 < |x-x_{0}| < δ, then |f(x)-L| < ε. |
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 (This experimental approach is modified from some as-yet unpublished material by Professor Bogdan Baishanski of Ohio State University.)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 φ, whose domain and range are of the form (0,a) and (0,b) respectively, where a and b are any members of (0,+∞]. |
Examples.
φ(t) = t, | considered as a function from (0,+∞) to (0,+∞) |
φ(t) = t, | considered as a function from (0,2) to (0,2) |
φ(t) = t^{2}, | considered as a function from (0,+∞) to (0,+∞) |
φ(t) = √t, | considered as a function from (0,+∞) to (0,+∞) |
φ(t) = t/(1+t), | considered as a function from (0,+∞) to (0,1) |
φ(t) = tan(t), | considered as a function from (0,π/2) to (0,+∞) |
φ(t) = 1-cos(t), | considered as a function from (0,π/2) to (0,1) |
I have chosen the name "funnel" to emphasize that the graph of φ(t) is narrow (i.e., φ(t) is small) for t near 0, like a kitchen funnel turned on its side; this is perhaps best illustrated by the funnel φ(t) = t^{2}. That property is not immediately evident in the graph of √t, which appears to be flat at its left end, rather than pointed. Nevertheless, √t is an increasing function, so √t is indeed a funnel.
Actually, the functions t, t^{2}, √t 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 φ is a funnel, with domain (0,a) and range (0,b). Suppose that 0 < a_{1} < a, and let b_{1} = φ(a_{1}). Then φ, considered as a function from (0,a_{1}) to (0,b_{1}), 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 φ
with the property that,
whenever x is a real number such that |x-x_{0}| is in the domain of φ, then x is in the domain of f and |f(x)-L| < φ(|x-x_{0}|). |
Of course, different limit assertions may require different funnels. For instance: