\[\lim_{(a,b)\to \infty}\int_{0}^{a}\int_{0}^{b}\sin\left(\frac{\pi x^2}{2}+\frac{\pi y^2}{2}\right)\, dy\; dx\]

Evaluate this limit, rounded to three significant figures.

If you come to the conclusion that this limit fails to exist, enter 666.

**Formal Definition**: \(\displaystyle \lim_{(a,b)\to\infty}f(a,b)=L\) means that for every \(\epsilon>0\) there exists an \(N\) such that \(|f(a,b)-L|<\epsilon\) whenever \(a>N\) and \(b>N\).

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