What is the limit of  tan(1/x) as x approaches infinity?

First, notice that as x approaches infinity, the inside term $\frac{1}{x}$ goes to zero.
Now, just evaluate tangent as $\frac{1}{x}$ goes to zero, $\tan \left(0\right) = 0$.
${\lim}_{x \to 0} \left(\tan \left(\frac{1}{x}\right)\right) = \tan \left({\lim}_{x \to 0} \left(\frac{1}{x}\right)\right) = \tan \left(0\right) = 0$.