Aunc.1961
net.math
utcsrgv!utzoo!decvax!duke!unc!bts
Sat Mar 13 20:28:32 1982
log(0) revisited
There is a problem with the argument that log 0 - log 0 = 0, unfortunately.
It is just as easy to take limits as follows:
Lim log(a) - Lim log(b) =
a -> 0 b -> 0 ???
There are many cases where you'll get into trouble if you assume that two
variables are going to 0 at the same rate. As far as ways to deal with log
of 0, I rather like infinitesimals. Replace the 0's with a and b, where
a and b are "infinitely close" to 0. Then the algebra works, but you won't
necessarily get 0 or even another infinitesimal as the answer. Adding a
single point at infinity looks nice topologically, but you get into trouble
using that point algebraically; it won't obey the same rules as "finite"
points. A good source for non-standard analysis is the first few chapters
of Stroyan and Luxemburg's text. (Oh yes, if you replace both occurrences
of 0 with the same infinitesimal, you do get 0.)
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