This subject is being made too complicated.

Division is defined as the inverse of multiplication.
  • If Quotient * X = 1, then Quotient = 1 / X
  • Now if X = 0, then Quotient * 0 = 1
  • What value can you assign to Quotient in this case?
The above indicates a serious problem with division by zero. Considering ordinary real numbers, there is no possible value for Quotient. End of discussion.

All the talk about infinity is more related to philosophy than mathematics. As far as I know, infinity is dealt with in set theory, which is more a branch of logic than a branch of mathematics.

I have not critically analyzed the posts to this thread, mainly because I do consider myself enough of an expert to avoid errors on this subject.

The above having been said, I would advise others to not trust anyone who posts here (including me) on this subject without verifying via a serious mathematical text.

For example, a very knowledgeable and intelligent individual once posted the statement that 1^infinity = 2.71828... (or the number usually referred to as "e"). This happens to be erroneous, which I only realized after doing some analysis. Due to my respect for the individual, I accepted his post until I had occasion to discuss it with a friend of mine who suggested that we try to verify the statement. This error has to do with the following very subtle concept.
  • Limit(1 + 1 / N ) approaches one as N grows without bound.
  • Limit[(1 + 1 / N)^N] approaches e (2.71828...) as N grows without bound.
  • It is invalid to claim that Limit(1 + 1 / N)^N equals Limit[Limit(1 + 1 / N)]^N as N grows without bound.
One must be very careful when replacing an expression with the limit of that expression. If anybody is interested, I can expand on this.

I wonder about the references to positive and negative infinity and left/right limits. It is not clear to me that transfinite numbers have signs. In complex analysis, infinity is a circle.

I do not think mathematicians make statements like 1 / 0 = infinity, instead they say 1 / X grows without bound as X approaches zero. Oddly enough, they sometimes assign a value to zero divided by zero, with one not necessarily being the value assigned. I do not remember any mathematical text which allows using a value like 1 / 0 as a real number in an equation.

As far as I know, mathematicians avoid treating infinity as a real number. The only literature I know of that deals with the concept uses the term "Transfinite Numbers" to refer to the cardinal number associated with certain sets. Once you open that can of worms, you are forced to accept more than one transfinite number. For example: The set containing all of the integers is smaller than the set containing all real numbers. In "slang" terms, there is more than one infinity.

I do not remember any literature which allows transfinite numbers to be used as equivalent to real numbers in analytical expressions.

I am not sure that you can find a reference which allows you to claim which transfinite number corresponds to Limit(1/X) as X approaches zero. Is it the transfinite number associated with the set of all integers, or is it the transfinite number associated with the set of all real numbers? If somebody can cite an authority for this, I will be interested to know of it. I would not be surprised if no such citation can be found. I will not be surprised if there is a citation. If there is a citation, I am guessing that it is the transfinite number associated with the set of all reals.