Is $(x+1)/(x^2-1)$ defined for $x=-1$? It might sound like a silly question, but I can't come up with a clear answer.
By looking at the expression, the answer should be "no", since $(-1)^2=1$ and we're in trouble. 
However, if I factorize: $(x^2-1) =(x+1)(x-1)$, $x=-1$ is still illegal. But now the terms cancel out and I am left with $1/(x-1)$, which is clearly defined for $x=-1$?
What happened? Was some information lost in the manipulation or was the original expression an "illusion"? I hope my question makes sense.

It is the exact same problem as saying:
Is $x/x$ defined for $x=0$ ?
The answer is no.
Even if you could change the formula with algebra to give it a value (which would obviously make sense), it is not defined per se.
