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