Numerical Methods: Euler to Runge–Kutta · Superchallenge problem

The backward Euler method is applied to $x' = y$, $y' = -x$ from $(1, 0)$, with $n$ steps of size $h = \dfrac{2\pi}{n}$ covering one period. Let $r_n = \sqrt{x_n^2 + y_n^2}$ be the final distance from the origin. Find $\displaystyle\lim_{n \to \infty} n\,(1 - r_n)$.