Limit can be defined by two axioms: 1. Strict inequality between limits implies, ultimately, strict inequality between functions. 2. For constant functions limit is trivial. How can basic results on convergence be derived from these axioms? In this paper we propose two answers: a) at the most elementary level- add two more axioms, b) at somewhat higher level, do it in three steps, and, in our forthcoming paper "Axiomatic definition of limit", a third answer- c) do it neater - in an abstract framework, where only order relations are present.
The currently used ǫ -δ definition of the limit appears to many beginners to be too difficult. In the rearrangement of the introductory material that we now propose, the epsilon-delta definition would be dethroned and would get the status of a technical result (see thms 7 and 8 below). The new definition is obtained by combining two well-known mathematical facts, which in current expositions appear as trivial consequences of the ǫ -δ definition. One of these facts (see (2) below) becomes now the fundamental property of the limit.
In this note we consider real-valued functions of a real variable and define lim f (x) as x tends to infinity. With obvious modifications one can phrase definitions in the cases x tends to c+, to c-, to -∞, and similarly define limit of a sequence of real numbers. Also, only minor modifications are necessary to define in these cases the extended limit (that can take as values real numbers, ∞ and -∞).
According to the ǫ -δ definition, limit is a relation between functions and numbers; according to our definition in this note, limit is a mapping from a class of real functions to real numbers. So we need now to define both the mapping and the maximal class on which that mapping can be defined so that the defining property (2) of the limit is satisfied. We do that in three steps:
we define limit of monotone bounded functions, 2) we define class of convergent functions (new meaning is attached now to the term convergent-later it becomes evident that it coincides with the traditional meaning),
we extend the definition of the limit to all convergent functions.
(It follows easily from theorems 7 and 8 that the class of convergent functions cannot be further extended with preservation of property (2)).
Is our definition more accessible and more intuitive than the epsilondelta definition? Before answering that question, the reader should consider possible graphical illustrations of definitions 1 and 2 below and compare them with graphical illustration of the ǫ -δ definition.
We start by considering functions which are ultimately monotone and bounded,i.e. we consider the class BM (∞) = {f | there exists x 0 such that f is bounded and monotone on the interval (x 0 , ∞)}.
is said to be a limit on BM (∞) if the following two conditions are satisfied:
Instead of L(f ) = λ we shall frequently write “lim f (x) = λ” or “f (x) → λ as x → λ " . In case f is increasing/decreasing we may write f (x) ր λ as x → ∞ , resp. f (x) ց λ as x → ∞.
Thm.1. There exists a limit L on BM (∞).
Proof. If f is bounded and increasing for x > a, we set L(f ) = sup f (x)|x > a, if f is bounded and decreasing for x > a, we set L(f ) = inf f (x)|x > a. We need only to verify that the condition (2) is satisfied. There are four cases to consider since f can be increasing or decreasing, and similarly g. We shall consider here only the case when f is decreasing for x > a ′ and g is increasing for x > a ′′ . (the other three cases are even simpler). Choose γ such that
Thm.2. There is only one limit L on BM (∞).
Proof. Suppose that there exist two limits, L ′ and L”, and a function f in
Using γ to denote also the constant function with value γ, we have by (1) that L ′ (γ) = L"(γ) = γ so from (3) it follows that L ′ (f ) < L ′ (γ) and L"(γ) < L"(f ), and therefore by (2) there exist a ′ and a" such that f (x) < γ for x > a ′ , and f (x) > γ for x > a". So, if x > max(a ′ , a"), we get f (x) < f (x), a contradiction.
Thm. 3. If f and g belong to BM (∞) and if there exists a such that f (x) ≤ g(x) for x > a, then L(f ) ≤ L(g).
Proof. Assume not true. Then L(f ) > L(g). By (2) it follows that there exists a ′ such that f (x) > g(x) for x > a ′ . Taking an x > max(a, a ′ ) we obtain a contradiction.
Thm. 4. Let N be a positive decreasing function on [a, ∞). Then (i) L(N ) = 0 if and only if (ii) for every positive integer n there exists x n > a such that N (x n ) < 1/n.
Proof. (i)-→(ii). Since L(N ) < L(1/n) we get from (2) that there exists c such that N (x) < 1/n for x > c.
(ii)→(i). Since N is decreasing on (a, ∞) we have that 0 < N (x) < 1/n for x > x n . By Thm. 3 and by (1) we get 0 ≤ L(N ) ≤ 1/n for every n. Thus L(N ) = 0.
The following three statements are equivalent:
Definition 2 . We say that the function f is convergent as x tends to ∞ if there exist a number a and functions m and M , both bounded and monotone on the interval [a, ∞) such that
We denote by C(∞) the class of functions f convergent as x tends to infinity. Thm. 5. The limit L on BM (∞) can be extended to the class C(∞), in such a way that the condition (2) remains satisfied.
and L(m") = L(M “) = L”, then L ′ = L". We derive from the assumptions above that m ′ (x) ≤ M “(x) for x > max(a ′ , a”). Thus by Thm. 3
It remains to be shown that the condition (2) is satisfied on C(∞). Let this time m ′ , M ′ , m", M " belong to BM (∞) and satisfy m
) we obtain from (2) and Theorem 1 that there exists a such that M ′ (x) < m"(x) for x > a.
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