Multi variables limit

Question

How do I solve a limit for two variables $$\displaystyle\lim_{x,y\to\infty}\dfrac{(xy)^{x^2}}{(x^2+y^2)^{x^2}}?$$

I don't know how to avoid having indetermination

Answer 1

Using the well-known inequality: $\dfrac{xy}{x^2+y^2} \le \dfrac{1}{2}$, this implies the limit to be $0$.