Limit $\lim_{x \rightarrow \infty, y \rightarrow \infty} \left( \frac{xy}{x^2 + y^2}\right)^{x^2} $ Given the followning limit:
$$ \lim_{x \rightarrow \infty, y \rightarrow \infty} \left( \frac{xy}{x^2 + y^2}\right)^{x^2} $$
To find limit I have made following steps:


*

*Let $ x = y $ ,then limit equals $0$

*Let $ x > y $ ,then consider the limit:


$$  \lim_{x \rightarrow \infty, y \rightarrow \infty} \left( \frac{x^2}{x^2 + y^2}\right)^{x^2} = \lim_{x \rightarrow \infty, y \rightarrow \infty} \left( \frac{1}{1 + \frac{y^2}{x^2}}\right)^{x^2} = 0$$ 
with respect to $$0 < y^2/x^2 < const$$


*Let $ y > x $ ,then consider the limit:


$$  \lim_{x \rightarrow \infty, y \rightarrow \infty} \left( \frac{x^2}{x^2 + y^2}\right)^{y^2} = \lim_{x \rightarrow \infty, y \rightarrow \infty} \left( \frac{1}{\frac{x^2}{y^2} + 1}\right)^{x^2} = 0$$ 
with respect to $$0 < x^2/y^2 < const$$
What could you say about my solution? 

Hint,
Assume $x = r \cos \theta$ and $y = r \sin \theta, \theta = constant$
Then, the limit changes to 
$$ \lim \limits_{r \rightarrow \infty}\left ( \dfrac {r^2 \sin \theta \cos \theta}{r^2}\right)^{r^2 \cos^2 \theta} \\ \Rightarrow \lim \limits_{r \rightarrow \infty} (\sin \theta\cos \theta)^{r^2 \cos^2\theta} = 0$$
As expression under bracket is $<1$
