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22 August 2003

Adding Infinities

by George E. Hrabovsky, President, MAST

A Continuation

Last week we explored what might be called the arithmetic of infinity.  Today we will continue by exploring the addition of infinity.

More Properties of Infinity

Last time we also introduced the idea of an indeterminate value.  It is important to understand that while the value of a function can be indeterminate, the value of a limit never is.  This is because a limit either exists or it does not, there is no chance that a limit may have more than one value.

Having said that I put forward the following two theorems for any limit L we have,

Theorem 1

∞ + L = ∞ .

Theorem 2

∞ + ∞ = ∞ .

The Math Challenge from Last Time

The math challenge was to prove the three corollaries.  The first of these is,  If f(x)  L≠0 as x  x_0 ∧ g(x) ∞ as x  x_0⟹ f(x) g(x)  ∞ as x  x_0.  The function 1/f(x)  L≠∞.  By  Theorem 1 from, "The Nitty Gritty," this implies that the function is bounded by some deleted neighborhood of x_0.  Since the function is so bounded we can substitute 1/f(x) for f(x) in Theorem 2 from, "Extreme Limits." This proves the corollary.
    The second corollary is, f(x)  L≠0 as x  x_0 ∧ g(x) ∞ as x  x_0⟹ f(x) /g(x)  ∞ as x  x_0. By Theorem 1 from, "Extreme Limits," we note that 1/g(x)  ∞ as x  x_0.  Then we apply Corollary 1 from, "Properties of Extreme Limits," (proven above) to prove the corollary.
    The third corollary is, f(x)  L≠∞ as x  x_0 ∧ g(x) ∞ as x  x_0⟹ f(x) /g(x)  0 as x  x_0.  By  Theorem 1 from, "The Nitty Gritty," this implies that the function is bounded by some deleted neighborhood of x_0.  Since the function is so bounded, then by Theorem 1 from, "Extreme Limits," the corollary is proved.

The Math Challenge

Can you prove the theorems of this column?

Math Resources for Limits

Online:

http://www.maths.abdn.ac.uk/~igc/tch/ma2001/notes/node37.html

This is a rather uninspired page, but it is the best one that I could find.

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 includes a very detailed discussion of infinite limits.

Created by Mathematica  (August 20, 2003)