# Limit It!

Calculus Level 5

Let $$x_1,x_2, \mbox{ and } x_3$$ be roots of the equation $$(x-1)(x-8)(x-31)=1$$

If $\lim\limits_{m\rightarrow\infty}\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{3}\cos^m\left(\pi n! x_i\right)=k+7$

for integers $$m$$ and $$n$$, find the value of $$k$$.

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