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Re: RE: Limits: Is there something I'm missing Here?
The answer to the last Ashraf's last question is mathematically rather uninteresting: in Mathematica Limit[f[x], x -> 0] *means* Limit[f[x], x -> 0, Direction -> -1]. As for the other points, well it is true that even the one sided limits of 1/x "do not exist" if a limit is required to be a real number. But it is often convenient to consider the "extended" real line with two additional points, called -Infinity and Infinity (one speaks of "compactifying" the real line, which then becomes topologically equivalent to a closed interval). This can be made perfectly rigorous, although of course the object thus obtained is no longer a field in the algebraic sense (in other words, you can't perform usual arithmetic with Infinity and -Infinity). Andrzej Kozlowski Yokohama, Japan http://www.mimuw.edu.pl/~akoz/ http://platon.c.u-tokyo.ac.jp/andrzej/ On Wednesday, February 12, 2003, at 05:52 PM, David Park wrote: > Ashraf, > > Strictly speaking you are correct. The limit does not exist. But there > is > such a thing as a one-sided limit, which is not a true limit but still > useful. > > In this case even the one-sided limits do not really exist because the > result is unbounded. But again, it is useful to signify this by saying > that > the limit is infinite. So it is rather loose language by both > mathematicians > and Mathematica. > > Perhaps one of the mathematicians in the group will give you a fuller > explanation. > > David Park > djmp at earthlink.net > http://home.earthlink.net/~djmp/ > > From: Ashraf El Ansary [mailto:Elansary at btopenworld.com] To: mathgroup at smc.vnet.net > To: mathgroup at smc.vnet.net > > Dear all, > One thing I've noticed that if we have a function which has two > different > limits (given two different directions) at one points , mathematica > would be > still give an answer though to my understanding the limit doesn't > exist in > such a case. > > Consider the following example: > a[x_]:=1/x > > Limit[a[x],x->0,Direction->+1] +Inf > > Limit[a[x],x->0,Direction->+1] -Inf > > Limit[a[x],x->0]. +Inf.... Maybe my calculus knowledge is a > bit > rusty but does the limit exist in this case?? > > > > Thank you > > > > > >