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:

1. Let $ x = y $ ,then limit equals $0$


2. 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 $$0 < y^2/x^2 < const$$

3. 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?