Infinite crossings

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

Image credit: Wolfram Alpha