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

Question

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.

Since $\tan x$ grows without bound at $\pi/2 + n\pi$ for $n \in \mathbb{Z}$, it doesn't make much sense to evaluate it at the rather arbitrary value of $10^{100}$ and draw any conclusions. — MT_, Apr 20 '14 at 18:52

score 6 · Accepted Answer · answered Apr 12 '14 at 22:53

6

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 $\pi$.

answered Apr 12 '14 at 22:53

It's actually not even defined for $(n+\frac12)\pi$, so the limit doesn't really make sense. – user2345215 Apr 12 '14 at 22:55
What about the limit of the sequence $a_n=\tan(n)/n$? – user85798 Apr 15 '14 at 21:17

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