Indeterminate FormsDate: 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. Thank you for your time. 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 number. You probably already know about this. Some operations can be 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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