What is so bad about the extended reals, they seem like a fine system of numbers to work with to me, you just have to remember the properties of oo and -oo.
Cheers, Adam "Chan-Ho Suh" <email@example.com> wrote in message news://3C099B3D.8A447554@math.ucla.edu... > > > Paul Lutus wrote: > > > "Chan-Ho Suh" <firstname.lastname@example.org> wrote in message > > news://3C082CE3.B54ECB73@math.ucla.edu... > > > > > Lutus is incorrect. > > > > Post your erroneous argument first. > > I have posted no erroneous argument, nor do I intend to ever do so. > > > > > > > > The equal sign refers to equality of numbers. > > [gibberish snipped] > > > > > > The example under discussion -- > > > > lim x -> 0, 1/x = infinity > > > > Oh, you're doing the "I'm wrong but don't want to admit it so I'll change the > discussion topic" approach. Note that 0 and infinity were switched in the > original post I responded to. > > > > > > -- is an unfortunate shorthand that appears to say: > > > > "as x approaches the limit of zero, 1/x equals infinity" > > > > -- which, of course, is false and is not what a limit represents. > > > > In the extended reals (one or two point compactifications) this makes perfect > sense. It is not the working mathematician's fault that some calculus teachers > are afraid to talk about the concept of points at infinity. > > > > > > > The left hand > > > side is a limit of (presumably) the real function 1/x as x approaches > > infinity and > > > so is precisely the real number 0.. > > > > You are looking at the wrong example. Read above. > > > > I'm looking at the example that started the thread. Do *you* know how to read? > > > > > > > The right hand side is (the way I am reading > > > it) the real number 0. Clearly 0 = 0. The thing to remember here is that > > the > > > limit operator returns a number (assuming the operator is well-defined and > > perhaps > > > assuming +/- infinity to be an 'extended' number). > > > > > > Lutus' mistake is common. > > > > Yours is much more common. You have overlooked this semantic ambiguity in > > the limit expression, the bane of math teachers. > > > > Yours is the most common mistake: taking one's limited knowledge to be > all-encompassing. This business of 'extended' reals is the bane of math > teachers who never understood the subject. > >