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11 July 2003

Limits By Themselves

by George E. Hrabovsky, President, MAST

A Continuation

Last week we explored the idea of a deleted neighborhood.  Today we will examine limits by themselves.

Theorems About Uniqueness

Uniqueness is a very important concept in mathematics.  A function must be unique at every point to exist.  The solution to an equation must be unique if it is to have any real usefulness.

Theorem 1:

If f(x) approaches any limit as x  x_0, then the limit is unique.

Theorems About Error

As some people have noted in the forum the idea of a limit is not exact, that is it is an approximation.  We talk of how a function approaches a value as a variable approaches another value.  This talk of approaching is vague and approximate to within the ϵ - δ values.  So we have an inherent error when we talk of limits.

 

Theorem 2:

The function f(x)  L as xx_0 if and only if f(x) = L + g(x) where g(x)  0 as x  x_0.

The Math Challenge from Last Time

The challenges this time are to prove the three theorems presented in the last column.  This first is, 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.  Let the limit of f(x) be L as x  x_0.  If we choose ϵ = 1, then we have δ such that 0 < | x - x_0 | < δ (the deleted neighborhood) implies that | f(x) - L | <1, and thus (by the triangle inequality)

| f(x) | - | L | ≤ | f(x) - L | <1,

or

FormBox[StyleBox[RowBox[{| f(x) |, , =, , RowBox[{|, L, |, , RowBox[{+, , 1.}]}]}], FontSize -> 18.], TraditionalForm]

Applying the triangle inequality again, this implies,

| f(x) | ≤ | L | + 1

in the deleted neighborhood 0 < | x - x_0 | < δ.  Thus we see that the function is bounded in the deleted neighborhood.  Thus the theorem is proved.

The second theorem to prove is 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.  Suppose L > 0, then let ϵ = L/2.  We can then choose δ such that 0 < | x - x_0 | < δ implies

| f (x) - L | <L/2 .

Then, by the interval definition of absolute values, we have

-L/2<f(x) - L<L/2,

or,

L/2<f(x) < (3L)/2,

thus f(x) > 0 when L > 0.  If we choose ϵ = -L/2>0, then (6) becomes

(3 L)/2<f(x) <L/2,

so that f(x) < 0, when L < 0.  Thus the theorem is proved.

The final theorem was 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.  Another way of saying this is that there are numbers N > 0 and FormBox[RowBox[{δ_1, >, , RowBox[{0, Cell[]}]}], TraditionalForm] such that f(x) >N so long as 0 < | x - x_0 | < δ_1.  Now, we also have the fact that any time ϵ > 0, we have δ_2> 0 such that g(x) > ϵ/N if 0 < | x - x_0 | < δ_2.  This implies that, given ϵ>0 we need only choose δ to be the minimum of δ_1 or δ_2 to get,

| f(x) g(x) | = | f(x) || g(x) | <N ϵ/N = ϵ

for all x such that 0 < | x - x_0 | < δ.  This is another way of writing f(x) g(x) 0 as xx_0.  Thus the theorem is proved.

The Math Challenge

Can you prove theorems 1 and 2 of this column?

Math Resources for Limits

Online:

http://www.geocities.com/pkving4math2tor1/1_lim_and_cont/1_01_04_one_sided_lim.html

This is a very nice page.

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 11, 2003)