That Is Close 3

Calculus Level 5

The sequence of real numbers \( \{ x_1, x_2, x_3, \ldots \} \) satisfies \[\begin{cases} \displaystyle \lim_{n \to \infty}(x_{2n} + x_{2n+1}) = 315 \\ \displaystyle \lim_{n \to \infty}(x_{2n} + x_{2n-1}) = 2003 \end{cases} \] Evaluate \(\displaystyle \lim_{n \to \infty}\dfrac{x_{2n}}{{x_{2n+1}}}\).

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