Date: Thu, 07 Oct 2000 21:17:19 -0400
From: Alex Bogomolny
Dear Mickey:
It's not a question whether it's bad or not, but whether it's useful.
Strictly speaking, the limit of 1/x as x goes to 0 does not exist at all - it can't be declared infinity or anything else. The limit from the right, i.e., when x is strictly positive, may be considered as infinite. The limit from the left, i.e. when x is negative, may be declared a negative infinity.
Assume you go ahead and declare the two one-sided limits as infinite. What do you gain? Infinity thus defined is not a number. Infinity times 0 is undefined. So think before you do that.
Do not also put all infinities considered in mathematics into a single basket. There are many; each treated differently and is used for a different purpose.
There's a point at infinity and a line at infinity, there are cardinal and ordinal infinities, and more. They are all very different.
All the best,
Alexander Bogomolny
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