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Adding Infinity

Date: 06/25/2003 at 11:33:13
From: Joshua Calais
Subject: Infinity

What is -infinity plus infinity?	

This question was our class discussion today in Calculus II. The 
discussion was whether it equals 0 or undefined. We believed it was 
zero, but didn't know how to come to a definite conclusion.

Date: 06/25/2003 at 12:19:04
From: Doctor Peterson
Subject: Re: Infinity

Hi, Joshua.

Infinity itself is undefined! It is not a number, so you can't add 
anything to it.

In a calculus class, you should be making a clear distinction between 
actual arithmetic and limits; we talk about infinity in the context 
of limits, where we are not really saying that we can add infinity 
and negative infinity, but rather that we can take the limit of the 
sum of two functions whose limits are infinity and negative infinity. 
In that setting, "infinity plus negative infinity" is indeterminate. 
This means that the limit of the sum depends on how the addends 
approach infinity. If one does so faster than the other, then it will 
govern the sum. We can't "determine" the limit of the sum without 
knowing the individual functions.

See these pages for more:

   Indeterminate Forms 

   Why Are 1^infinity, infinity^0, and 0^0 Indeterminate Forms? 

Note that these are called "indeterminate FORMS," not values. We 
don't actually treat infinity as a value when we do this; it is only 
the "form" of the limit that looks like "infinity minus infinity," 
and we have to change its form in order to evaluate it correctly.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Calculus
High School Number Theory

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