Can someone double-check my work here? Thanks, if you can.

$\displaystyle

\lim_{x\to\infty}[x(1-cos(1/x))]

$

Put x to the denominator to use L'Hopital's rule, etc. Applied chain rules, etc.

$\displaystyle

\lim_{x\to\infty}[sin(1/x)]

$

Thus, if x goes to infinity, the above is sin(0) = 0. Final Answer: Zero