For example, does the limit of $f(x,y) = \frac{bxy}{xy}$ for any constant $b$ exist for $(x,y) \to (0,0)$?
Does the fact that for $x=0$ and $y=0$ you have a problem with deviding by zero imply that there is no limit? Or could you extend the definition of a limit with "for all $x$ and $y$ in the domain of $f$..."?
In fact, this question arose from the following question:
For which constants $a$, $b$ and $c$ does the limit $(x,y) \to (0,0)$ exist for $f(x,y) = \frac{ax^2+bxy+cy^2}{xy}$?
When approaching $(0,0)$ through lines $y=mx$, it turns out that the limit depends on $m$ if not both $a$ and $c$ are $0$. So the first condition for the existing of a limit is that both $a$ and $c$ are $0$.
That leaves the question for which $b$ the limit $(x,y) \to (0,0)$ exist for $f(x,y) = \frac{bxy}{xy}$. And this is where we started doubting the solution. We found 3 approaches, which we don't know is the right one. Can you simply say:
Hey, $f(x,y) = \frac{bxy}{xy}$ simply results to $b$, so the limit exists for every $b$ (and equals $b$)? Or should you say:
$f(x,y) = b$ for $x \ne 0$ and $y \ne 0$, and $f(x,y)$ is undefined for $x = 0$ or $y = 0$. And this makes the limit nonexistent, because for pairs $(x,y)$, even though close to $(0,0)$, you cannot guarantee anything about $|f(x,y)-b|$ because if either $x = 0$ or $y = 0$ the function isn't even defined properly. Or is it the case that:
$f(x,y) = b$ for $x \ne 0$ and $y \ne 0$, and $f(x,y)$ is undefined for $x = 0$ or $y = 0$. And this is no problem, the limit still exists for every $b$ (and is $b$)?