An Archimedean spiral starting at the origin turns counterclockwise and has its first intersection with $y = 0$ at $x = -\pi$ . The spiral satisfies

$x^{2}+y^{2} = f \left(\frac{x}{y} \right).$

Find $\displaystyle \lim_{z \rightarrow 0} f(z)$ .

The answer is 2.4674.

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The equation of the spiral is $r=\theta$ .

Therefore, the parametrized equation in cartesian plane is:

Then:

So:

Therefore its limit is $(\arctan\pm\infty)^2=\left(\pm\frac\pi2\right)^2=\frac{\pi^2}4$ .