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Indeterminate Forms

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Date: 09/18/97 at 13:24:37
From: White, Pam
Subject: Infinity

Dear Dr. Math,

We were sitting in Calculus class one day, when a question
arose. What is infinity divided by infinity?  If you have an answer,
we would appreciate an explanation also.

Beebe High School Calculus Class
```

```
Date: 09/18/97 at 15:21:58
From: Doctor Rob
Subject: Re: Infinity

This is an excellent question.

You cannot actually do arithmetic with "infinity" as if it were a
defined quite well, but you run into trouble with others.

I guess you must mean by "infinity divided by infinity" something of
the form lim(x->a)[f(x)/g(x)], where lim(x->a)[f(x)] = infinity and
lim(x->a)[g(x)] = infinity. As an example, you might have a = 0,
f(x) = 1/x^2, and g(x) = c/x^2. Then the quotient is a constant c, and
the limit as x->0 is c. On the other hand, g(x) = 1/x is also
possible, and the quotient is 1/x, whose limit as x->0 is infinity.
Yet another possibility is g(x) = 1/x^3, so the quotient is x, whose
limit as x->0 is 0.  Thus the limit of the quotient can be any real
number or infinity.

The result of this situation is that it is very hard to say what
"infinity divided by infinity" is. In fact, mathematicians call this
one of the "indeterminate forms." There are seven of these. The others
are "0/0", "0*infinity", "infinity - infinity", "1^infinity", "0^0",
and "infinity^0".  For more about them, see:

http://iq.orst.edu/mathsg/SandS/lHopital/index.html

-Doctor Rob,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Analysis
High School Number Theory

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