What is the difference between $f(x) = \dfrac{(x + 2)(x - 2)}{x (x - 2)}$ and $g(x) = \dfrac{x + 2}{x}$? Can you always cancel the (x - 2) factor? Isn't an asymptote lost when that is done? When I evaluate it at $x=2$ with a computer, like Wolfram Alpha or something, the software always seems to cancel and then evaluate. That's why I'm confused. I would think it should be undefined.
Edit: Actually, I should have used the term "undefined point" rather than "asymptote". What I'm actually interested in is the derivative. Does the derivative have to preserve this undefined point or not? It seems to me that I should preserve the $(x-2)$ when calculating the derivative in order to preserve the undefined point, but the computer invariably cancels it.