I'm reading Adams' Calculus - A Complete Course and got stuck on something I'm guessing is quite easy. Anyway, I'm wondering why it is that the limit
$$ \lim_{x\to 2}\frac{x-3}{(x-2)(x+2)} $$ does not exist, whereas $$ \lim_{x\to 2}\frac{x-3}{(x-2)^2} $$ does exist (equal to $-\infty$).
What is it about $(x-2)^2$ that enables its existence?
On
Actually, both limits do not exist. However, some books (and for valid reasons) accept infinity as an answer. The difference in your examples is that in the first case, the left limit and the rigth limit are not the same (pos and neg infinity), whereas in the second example, both right and left limit are the same: pos infinity. If the right and left limit do not yield the same answer, the limit does not exist. Graph both functions to see what is going on at their asymptotes
The function $(x-2)^2$ tends toward $0$ from above as $x\to 2$, while $(x+2)(x-2)$ approaches $0$ from above or below depending on which direction $x$ approaches $0$ from. Left- and right-hand limits must agree for limits to make sense.