>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?
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.