How to explain this result about $\pi$ and $\arctan$ function?

Question

Let $x$ denotes an integer such that $x > 1$.
We define the function $f$ such that:
$$f(x)=\frac{1}{\pi}\arctan(x)$$ We have: $$f(x)=\cfrac{1}{a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+...}}}}$$ ($a, b, c, d$ are integers $\geq 1$) I want to prove that: $$\lim_{x \to \infty} \frac{x}{b} \approx 1.27$$ For example with $x=1000$ we have: $$f(1000)=\cfrac{1}{2+\cfrac{1}{784+\cfrac{1}{1+\cfrac{1}{8+...}}}}$$ And $$\frac{1000}{784} \approx 1.27$$

Thanks.

Answer 1

Hint: $$\frac\pi2-\arctan(x)=\arctan\left(\frac1x\right)\sim \frac 1{ x}.$$ as $x\to\infty.$

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Addition: You'll get $b$ is the nearest integer to $\frac{\pi x}4-1.$ So the limit of $\frac xb$ is $\frac{4}{\pi}\approx 1.27324.$