Some Thoughts about Limits

Abstract

This brings us to the purpose of this note, which is twofold. A first objective is to simplify the formulation and proof of theorems on limits. A second is to give precise definitions to, and so legitimize, certain modes of expression that are avoided by algebraists but embraced by analysts. There is sense in which theorems on limits are much easier than many of the standard homework problems in elementary calculus. For example, I would defend the view that it's easier to prove the theorem about the limit of sum than to integrate (sec x)5. Yet many people have the opposite perception, as suggested by the anecdote above. Why is this? Although various answers can be given, certainly good deal of the trouble is caused by piling up of logical terms: For any E there exists 8 such that all x of certain kind something happens.... Quantifiers are the quicksand in which the path ends. Here we give formulation in which quantifiers are kept separate, so that not too many of them pile up in one place. The approach has been classroom tested at UCLA, and the improvement in student performance is wholly out of proportion to the modest mathematical content of the innovation. The latter consists solely in the formulation and use of Theorem 1 (which follows). Turning to the second objective, there seems to be consensus that certain modes of expression common in analysis should be kept below stairs, in the scullery, and not admitted to polite company. As an extreme case, I recall reading somewhere that we musn't say function is increasing because, after all, it's one and the same function all the time, so can't increase (or decrease or change in any other way). Still less, then, can one use phrases like for x near a or for large x. The view taken here is that such phrases have an honorable tradition in analysis, and express concepts of great importance, and should be used. If you want to say that l/x is bounded near x = 1, or is less than 0.001 large x, go ahead and say it. All that's missing is precise definition. And not even that will be missing if you read far enough.