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Topic: Is (t^2-9)/(t-3) defined at t=3?
Replies: 166   Last Post: Oct 30, 2013 9:41 AM

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Peter Percival

Posts: 1,306
Registered: 10/25/10
Re: Is (t^2-9)/(t-3) defined at t=3?
Posted: Sep 29, 2013 3:05 AM
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Hetware wrote:
> [...] If I
> choose to interpret (t-3)/(t-3) = 1 for all real numbers, I don't see
> how it results in a logical contradiction.


If (t-3)/(t-3) = 1 holds for all real numbers, then (considering the
case t = 3) leads to the conclusion that 0/0 = 1. But if / is defined
(on the reals, say) so that x/y = z iff x = yz, then 0/0 if it means
anything at all can mean all and any real numbers. Since definition is
supposed to fix the meanings of symbols not leave them open to meaning
everything and anything, 0/0 hasn't been defined. For which reason the
proviso y =/= 0 accompanies x/y = z iff x = yz. Since 0/0 hasn't been
defined it can't be 1, therefore it isn't true that (t-3)/(t-3) = 1
holds for all real numbers.

Can /some/ meaning be given to 0/0 (outside the reals perhaps?) Maybe,
but that won't help with real functions like your F.

--
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9/28/13
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Peter Percival
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