Limits and Sequences

  • If you always put limit on everything you do, physical or anything else. It will spread into your work and into your life. There are no limits. There are only plateaus, and you must not stay there, you must go beyond them. (Bruce Lee)
  • The mind is the limit. As long as the mind can envision the fact that you can do something, you can do it, as long as you really believe 100 percent. (Arnold Schwarzenegger)
  • After experience had taught me that all the usual surroundings of social life are vain and futile; seeing that none of the objects of my fears contained in themselves anything either good or bad, except in so far as the mind is affected by them, I finally resolved to inquire whether there might be some realgood having power to communicate itself, which would affect themind singly, to the exclusion of all else: whether, in fact, there might be anything of which the discovery and attainment wouldenable me to enjoy continuous, supreme, and unending happiness. (Spinoza)
  • Sapiens nihil affirmat quod non probate. 

Alternative Definitions of Continuity and Derivative

In Calculus books you usually find definitions of continuity and derivative based on the notion of limit: A function x:R\rightarrow R is said to be continuous at t if

  • x(t)=\lim_{dt\rightarrow 0}x(t+dt),

and differentiable at t with derivative \dot x(t) if

\dot x(t)=\lim_{dt\rightarrow 0}\frac{x(t+dt)-x(t)}{dt}\quad \mbox{where } dt\neq 0.

If you define continuity and derivative this way using limits, obviously there is a reason to confront the student with the (difficult) concept of limit.

The notion of limit relates to converging sequences with \lim_{dt\rightarrow 0^+}x(t+dt)=x(t) expressing that for any decreasing sequence of positive time steps dt_1>dt_2>....>dt_n>dt_{n+1},.... approaching 0, the difference \vert x(t+dt_n)-x(t)\vert is smaller than any given positive number if only dt_n is small enough (but not zero).

The notion is extended to also dt_n<0 with \vert dt_n\vert approaching 0.

In mathematical terms this is usually expressed as: For any given \epsilon >0 there is a \delta >0 such that

  • \vert x(t+dt)-x(t)\vert <\epsilon \quad\mbox{if }\vert dt\vert <\delta .

This looks more precise or “mathematical”, but if you do not relate \delta to \epsilon, then it is as vague as saying that x(t+dt) is close to x(t) if dt is small, which is a purely qualitative statement.

Quantitative vs Qualitative Definitions

On the other hand, in our definitions of Lipschitz continuity and differentiability, no limits are visible.

You can argue that our definitions are more precise since they are quantitative, not just qualitative, as expressed through the Lipschitz constant L and the constant C_u.

Is it good or bad? Are we missing something using this approach? Well, you may judge yourself? Does the notion of limit capture the essence of continuity and differentiability?

Computational mathematics without limits.

An answer may be suggested by Achilles and the tortoise: With the limit/sequence definition, Achilles can be seen approaching the tortoise in a seemingly neverending sequential limit process with always half of the rest remaining, which appears paradoxical. With the definition without limit/sequence, Achilles will simply at a certain moment in time have traveled the same distance as the tortoise and thereafter be ahead: No paradox.

In fact, the limit/sequence definitions, seemingly requiring that mysterious inifinitely small yet nonzero quantity dt, have created a lot of confusion and trouble through the history of Calculus, for both teachers and students, trouble which serves no reasonable purpose. To require that something locally is close to a constant or linear function, which we do in our limit/sequence-less definitions of continuity and differentiability, does not invite to any paradoxes, real or imagined.

The limit/sequence definitions, commonly viewed to be too difficult for high-school, form the core of university Honors Calculus with the pretention of giving a deeper understanding.

Sequences from Computation

In our approach, sequences will naturally occur as the output of a computational algorithm which generates a sequence u_1,u_2,..., such as fixed point iteration/Newton’s method, or by successive reduction of the time step k with a factor 2.

We will meet sequences satisfying \vert u_n-u_{n+1}\vert\le 10^{-n}, which uniquely determine a decimal expansion of a unique number. But we see no reason to consider other sequences than those generated by such computational algoritms. This simplifies the mathematics without loss of anything essential.

To Think About

  • What is the use of limits in a standard Calculus text?
  • What sequences in a standard Calculus text arise naturally?

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