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new new math limit
Posted:
Sep 16, 1998 8:37 PM


>Subject: Re: new new math limit > I> recently saw a course objective for first semester calculus that >suggested that the student will learn to find the limit of a function >defined by a table, a graph or an analytical expression.
>I practically swallowed my tongue. Anybody else see a problem with this?
>Geoff H
From a rigorous point of view: Weierstrass' definition of limit, universally adopted because limits so defined are UNIQUE, hence correct, is
Any function f(x) claiming to have a limit as x approaches a must provide
i) a suggested guess L for a limiting number and
ii) an argument proving that for any e > 0, there is some delta(e) > 0 such that
for any x in f's domain,
IF x does not = a and a  delta(e) < x < a + delta(e)
THEN L  e < f(x) < L + e.
If a function is defined by a table, its domain is finite, so any guess `L' formally works, since delta(e) can be taken small enough so that there are NO x in f's domain satisfying x not = a and a  delta(e) < x < a + delta(e) .
The logical IF.....THEN..... statement is formally true, since IF P THEN Q is considered true if P is false by the rules of logic.
Thus, using Weierstrass's definition, such a function has a `limit'.
This does not coincide with our intuitive sense of the concept. Thus the question of a function having a limit a x > a (in the accepted mathematical sense) should be confined to values of a that are cluster points of the domain of f; in beginning calculus such would be either in an open interval or at an endpoint of an interval in f's domain.
Thus the question of a tabular function or a graph having a `limit' should not arise; such suggestions take us back to the vagueness that characterized calculus from the time of Newton and Leibnitz until Cantor, Dedekind, Cauchy, Weierstrass et al. resolved the problem.
The question makes no (welldefined) sense.
H. Swann
San Jose State U.



