Limit on two variables approaching infinity

Question

I have a question on the limit of $$\lim_{x,y\to\infty}\frac{(x-1)(y-1)}{xy}$$ I had a look on answers and theory like the following question: Limit question as $x$ and $y$ approach infinity?

So if I'm getting it right, the limit must exist by approaching by any path, that is, if we make $y=x$

$$\lim_{x\to\infty}\frac{(x-1)^2}{x^2}=1$$

which also holds for $y=x^2$, but not for things like $y=x^{-2}$:

$$\lim_{x,y\to\infty}{x(x-1)(x^{-2}-1)}=-\infty$$

and thus the limit doesn't exist. Am I getting it right?

Thanks for your help!

Answer 1

se that $$\frac{(x-1)(y-1)}{xy}=\frac{xy-y-x+1}{xy}=1-\frac{1}{x}-\frac{1}{y}+\frac{1}{xy}$$

Answer 2

HINT:

$$\lim_{x,y\to\infty}\frac{(x-1)(y-1)}{xy} = \lim_{x,y\to\infty}{(1-\frac1x)(1-\frac1y)}{}$$

Answer 3

It is unclear exactly what $\lim\limits_{x,y\to\infty}f(x,y)$ means.

Your comments are correct for $$ \lim_{x^2+y^2\to\infty}f(x,y) $$ However, one might also consider the limit $$ \lim_{\min(x,y)\to\infty}f(x,y) $$ for which your argument does not work.