Infinity over Infinity
Date: 4/10/96 at 18:50:50 From: "Jeremy Vigneault" Subject: Infinity question Dear Dr. Math, I have a question maybe you can answer. My electronics teacher and I are having a disagreement. He says that Infinity divided by Infinity equals one. I most certainly disagree; I say that infinity over infinity is indeterminate, because infinity is a concept, not a finite number. Could you please help me in this discrepancy?
Date: 4/12/96 at 21:36:13 From: Doctor Syd Subject: Re: Infinity question Dear Jeremy, Good work! You are correct. Many people are confused by infinity; you are right that is a concept and not a number the way that 28 is a number. There are sort of some different "sizes" of infinities, so this means that a quotient that looks like infinity over infinity can sometimes be a real number, and sometimes it is just infinity. Maybe to prove your case to your teacher you could think about the following problem: What is the limit as x approaches infinity of the expression x^2/(x-5)? If you "plug in" infinity for x in this expression you getinfinity over infinity, but if you apply L'Hopital's Rule, you see the the numerator dominates the denominator (you can see this without L'Hopital's Rule, too: x^2 gets big a lot faster than x does, right?), so the limit as x approaches infinity will be infinity, not one. Good question! Hope this helped some. -Doctor Syd, The Math Forum
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