About double integral convergence Does the below integral converge? 
$$\lim_{n\to \infty}\int_{|x|<n} \int_{|y|<n}\sin\left(x^2+y^2\right)\,dx\,dy$$
I saw this on the multiple choice test. The choices were
$$0 \\ \pi\\ -\pi\\ \text{does not converge}$$
I am sorry to ask this question, but I got stuck on this. Thanks in advance for any ideas.

\begin{align}
&\lim_{n\rightarrow \infty}\int_{-n}^n{\int_{-n}^n{\sin \left( x^2+y^2 \right) \text{d}x\text{d}y}}\\&=\lim_{n\rightarrow \infty}\int_{-n}^n{\int_{-n}^n{\left[ \sin x^2\cos y^2+\sin y^2\cos x^2 \right] \text{d}x\text{d}y}}\\
&=2\lim_{n\rightarrow \infty}\int_{-n}^n{\int_{-n}^n{\left[ \sin x^2\cos y^2 \right] \text{d}x\text{d}y}}\\
&=2\lim_{n\rightarrow \infty}\int_{-n}^n{\sin x^2\text{d}x}\int_{-n}^n{\cos y^2\text{d}y}\\
&=2\int_{-\infty}^{\infty}{\sin x^2\text{d}x}\int_{-\infty}^{\infty}{\cos y^2\text{d}y}\\
&=2\sqrt{\frac{\pi}{2}}\sqrt{\frac{\pi}{2}}=\pi 
\end{align}
