limits - Evaluating $limlimits_{xmathoptoinfty}frac{tan x}{x}$

I need to find

$$\lim\limits_{x\mathop\to\infty}\frac{\tan x}{x}$$

For some reason mathematica just returns my input without evaluating it.

For what it's worth, $\dfrac{\tan(10^{100})}{10^{100}}\approx -4\times10^{-101}$, so the limit is probably $0$. (...)

I'm guessing this has been asked before but I can't find it.

Answer

The limit does not exist: Since the tangent function has poles at every point of the form $\left(n + \frac 1 2\right) \pi$, the quantity

$$\frac{\tan x}{x}$$

is unbounded on every interval of length greater than $\pi$.