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? 

For $x, y >1$, we have the fact that
\begin{align}
2\leq\frac{x}{y}+\frac{y}{x}
\end{align}
which means
\begin{align}
\left(\frac{xy}{x^2+y^2}\right)^{x^2}=\left(\frac{1}{\frac{x}{y}+\frac{y}{x}}\right)^{x^2} \leq \left(\frac{1}{2}\right)^{x^2}.
\end{align}
