I have to evaluate the following without L'Hopital's rule

$\underset{x\to \mathrm{\infty}}{lim}x\mathrm{tan}(1/x)$

I can simplify this to be

$\underset{x\to \mathrm{\infty}}{lim}x\mathrm{sin}(1/x)$

because

$\underset{x\to \mathrm{\infty}}{lim}\mathrm{cos}(1/x)=1$

However, after that, I'm totally lost. L'Hopital's rule seems like my only option. Can someone help me out?

$\underset{x\to \mathrm{\infty}}{lim}x\mathrm{tan}(1/x)$

I can simplify this to be

$\underset{x\to \mathrm{\infty}}{lim}x\mathrm{sin}(1/x)$

because

$\underset{x\to \mathrm{\infty}}{lim}\mathrm{cos}(1/x)=1$

However, after that, I'm totally lost. L'Hopital's rule seems like my only option. Can someone help me out?