# Infinite crossings

Calculus Level 4

The graph of $$f(x) = sin(\frac 1 x)$$ gets a little...crazy near $$x=0$$. Specifically, the graph of $$f(x)$$ has an infinite number of zero-crossings (roots, or values of $$x$$ such that $$f(x) = 0$$) between the first one at $$x=\frac{1}{\pi}$$ and $$x=0$$.

Let $$x_n$$ be the $$n^{th}$$ zero-crossing of the function $$f(x)$$, where $$x_1 = \dfrac{1}{\pi}$$, and $$x_2 = \dfrac{1}{2\pi}$$ and so on in the negative direction. Also let $$f'(x)=\dfrac{df(x)}{dx}$$.

Find $$\displaystyle \lim_{n \to \infty} \frac{f'(x_{n+1})}{f'(x_n)}$$

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