The function $cos(2sin(\frac{1}{x}))$ is just as bizarre as the $sin(\frac{1}{x})$ example we saw in class: it oscillates infinitely often between $cos(2)$~~$-0\mathrm{.}46$ and $1$ as $x$ approaches zero $($why$?).$ Here is a graph of this function on $-1,1-$ you can see how my graphing utility starts to break down close to $x=0.$

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$($a$).[4]$ Determine $\underset{x0}{lim}{x}^{2}cos(2sin(\frac{1}{x})),$ with justification.

$($b$)[1]$ Use your favourite graphing utility to plot $x^{2}cos(2sin(\frac{1}{x}))$ on $-1,1($doesn$\text{'}$t it look neat?$)$ and on $-0\mathrm{.}1,0\mathrm{.}1.$ Is the picture compatible with your answer from part $($a$)?$ Briefly explain why or why not.

$($c$)[2]$ What is $\underset{x}{lim}cos(2sin(\frac{1}{x}))?$