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04 July 2003

The Nitty Gritty

by George E. Hrabovsky, President, MAST

Okay, We Have To Do This

I have tried to figure out a way of motivating this by example, but I have been unable to cobble together anything that is not terribly contrived.  Up to now we have treated limits as though they were something awesome and monolithic.  In reality, as interesting as they are in their own light, for us they are only a means to an end.  To get to that end, we need to understand how to use and combine limits in different ways.  Rather than breeze through this subject by giving you a list of theorems, we will break them up into digestible chunks to help you understand the theorems more clearly.  These next few columns will, by necessity, be a little more mathematical than many others we have done recently.

Deleted Neighborhoods

Any interval of the form

a < x <b

where (a, b) form a neighborhood, is called a deleted neighborhood as the midpoint of the neighborhood is not included in either of the subsets of the interval (a, x) or (x, b).FormBox[StyleBox[Cell[], FontSize -> 18], TraditionalForm]

Theorem 1:

If f(x) approaches any limit as x  x_0, then we can say that f(x) is bounded in some deleted neighborhood of x_0 .

Theorem 2:

If f(x) has a nonzero limit L at x_0, then we can say that there exists a deleted neighborhood of FormBox[RowBox[{x_0, Cell[TextData[]]}], TraditionalForm]such that f(x) has the same sign as L.

Theorem 3:

If f(x) is bounded in a deleted neighborhood of x_0 and if another function, g(x), approaches 0 as xx_0, then we say that f(x) g(x) 0 as xx_0.

The Math Challenge from Last Time

The challenge was to prove that for a limit to exist its right and left limits must exist and they must be the same.  Let's assume that we have a function that has a limit, L.  Then, by the definition of a limit, given ϵ > 0 there exists some δ > 0 such that | f(x) - L | < ϵ for all x such that 0 < | x - x_0 | < δ.  From our exploration of absolute values we know that this implies that the limit exists for either 0 < x - x_0 < δ or 0 < x_0 - x < δ, hence both the right and left limits exist and they are equal to L.
    Now suppose that the right and left limits exist and are equal.  Then, by the definition of a limit, given ϵ > 0 there exists some δ > 0 such that | f(x) - L | < ϵ for all x such that 0 < x - x_0 < δ or 0 < x_0 - x < δ.  We know that this implies that the limit exists for 0 < | x - x_0 | < δ, thus the limit exists and is equal to L.  Thus for a limit to exist both the left and right limits must exist and be equal.

The Math Challenge

Can you prove theorems 1, 2, and 3?

Math Resources for One-Sided Limits

Online:

http://www.math.montana.edu/frankw/ccp/calculus/estlimit/onesided/learn.htm

This is a very simple page that might provide some basics to help in understanding.

For a more advanced treatment go here,

http://www.maths.abdn.ac.uk/~igc/tch/ma2001/notes/node35.html

Books:

Richard A. Silverman, 1969, Modern Calculus and Analytic Geometry, MacMillan Company, New York (Dover Publications has reprinted this book with corrections in 2002). This has a very nice chapter on limits that include detailed discussions of one-sided limits and deleted neighborhoods.

Created by Mathematica  (July 3, 2003)