I agree that not all infinities are equal, but your example does not
necessarily lead to the problem you state. If zero is *defined* as being
1/Infinity, then f(x) = 1/(1 -1) = 1/0 = Infinity, as you state. However,
g(x) = (x - 1)(x + 1)
g(x) = (1 - 1)(1 + 1) = 0 * 2 = 2 * Infinity, so
g(x) * f(x) = (2 * Infinity) * (1/Infinity) = 2 * (Infinity/Infinity) = 2
Oops! Now it's my turn to get out the rust.
g(x) = (x - 1)(x + 1)
g(x) = (1 - 1)(1 + 1) = 0
* 2 = 2 * (1/Infinity), so
g(x) * f(x) =
(2/Infinity) * (Infinity) = 2 * (Infinity/Infinity) = 2
Got my "f 'n g's
mixed up" :-(
The statement that g(x) approaches infinity as x approaches 1, although it is
certainly true, does not clearly state whether and how this statement defines
zero. Is it 1/Infinity, 2/Infinity? Do you see what I am saying? The line of
reasoning you give does the same thing that all other lines of reasoning which
prove that 2 = 1 do that I have seen; it *redefines* zero or infinity in the
middle of the line of reasoning without actually making the existence of that
redefinition clear. Since the individual following this line of reasoning does
not notice this shift from one definition of zero to another, he naturally
concludes that there is a problem with defining zero as being 1/Infinity, when
the actual problem is that a term, namely zero, has been *redefined* in the
middle of the line of reasoning, something which is virtually guaranteed to
lead to faulty conclusion.
Phil
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