# How can functions disagree with the values of its expansions at some points on an algebraic curve

I found a curve, in which some function has at least two expressions, which differ infinitely much!! Is there any error in the thoughts?
The curve is defined by

"$$Ax²+Bx+C=u²,$$ $$A'x²+B'x+C'=v²,$$
provided either $A$ and $A'$, or $C$and $C'$, are squares; it may be assumed that the left-hand sides have no common zero on the projective straight line, since otherwise this would define a curve of genus 0."
(Changed from the book Number theory, An approach through history from Hammurapi to Legendre by A.Weil)

Further define four points $P_{+,+}$,$P_{+,-}$,$P_{-,+}$,$P_{-,-}$, respectively by $(0,1,a,a')$,$(0,1,a,-a')$,$(0,1,-a,a')$,$(0,1,-a,-a')$. Now we consider the function $f$=$a'u+av-t_0$, with $t_0$ given by $(A'B-AB')/(2aa')$. By (I), we have the expression for $f$:

Since $(a'u+av)(a'u-av)=(A'B-AB')x+(A'C-AC')$, by dint of some easy calculations,
$f=((A'B-AB')x+(A'C-AC'))/(a'u-av)$.

Now, substituting the values previously defined, we find that $f$ should actually go to infinity at $P_{+,+}$, and at $P_{-,-}$. Nevertheless, direct substitution tells us that $f$ does not go to infinity at these points!! How can this be true?
Ever since the times of the high school, the identities of different expressions for functions have always been a useful (if not the only) means of obtaining important results (to me). But I seem to fail to manipulate correctly here. One possible explanation for this phenomenon IMO is in the process of embedding the curve in the projective line, but I figured not out where exactly it went wrong.

Hope the formulation is not too confusing or obscure.

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Personally, I got bored reading your introduction, which doesn't contain any interesting information, before I even got to the question. In my opinion, you are more likely yo get people to read your question if you stick to mathematics (and use latex). – Alex B. Aug 16 '11 at 8:01
Like Alex B. said. – Did Aug 16 '11 at 8:04
Thank you for giving me the suggestions; I will try to emend. – awllower Aug 16 '11 at 10:55
There is no introduction now; if it is concerned with the latex, please teach a young boy who knows little about latex. Sorry for any inconvenience; I just want to know the answer, and thank you all. – awllower Sep 1 '11 at 13:18