HMMT 2003 K4

The sequence of real numbers $x_{1}, x_{2}, x_{3}, \ldots$ satisfies $\lim _{n \rightarrow \infty}\left(x_{2 n}+x_{2 n+1}\right)=315$ and $\lim _{n \rightarrow \infty}\left(x_{2 n}+x_{2 n-1}\right)=2003$. Evaluate $\lim _{n \rightarrow \infty}\left(x_{2 n} / x_{2 n+1}\right)$.