Tips to solve limits of 2 variables

Question

Anyone got a tips on solving this type of limits $$ \lim_{(x^2 + y^2) \to \infty} \frac{e^{x^2+xy+y^2}-1}{e^{x^2+y^2}-1}$$ I'm always having a trouble finding a good method to solve for limits of 2 variables approaching infinity, usually I'd either use Taylor (mostly if limit approaches 0) or Polar coordinates, but I tried both here without any results, what is the way to think here?

Answer 1

You can rewrite the expression as $$ \frac{e^{xy}-e^{-x^2-y^2}}{1-e^{-x^2-y^2}}. $$ The second term in both the numerator and denominator goes to $0$ as $x^2+y^2\to\infty$. But now we can see the problem, which is that the behaviour of the numerator can change widly depending on the signs of $x$ and $y$.

Namely, if you take $y=-x$, for instance, you now have $$ \frac{e^{-x^2}-e^{-2x^2}}{1-e^{-2x^2}}\xrightarrow[x\to\infty]{}0. $$ But if $y=x$, you have $$ \frac{e^{x^2}-e^{-2x^2}}{1-e^{-2x^2}}\xrightarrow[x\to\infty]{}\infty. $$ Another possibility, mentioned in the comments, is to also consider $y=0$, and now the expression is $1$. All in all, the limit cannot exist.