# What is the limit $( 1 + 1/\tan(n) )^{\tan(n)}$

What is the result of $\displaystyle \lim_{n\to \infty}\left(1+\frac{1}{\tan(n)}\right)^{\tan(n)}$ = ?
The limit does not exist as stated by Adam Rubinson.

Looks like the requested answer is e but then the question is wrong.
What is the correct question to give the answer e?

Did answered correctly. But I think this is some kind of typographical error.
This question appeared on a high school test and the intended question should have a small error.
What is the correct question to give the answer e changing only one value?

• What makes you think that this limit exists? – us2012 Aug 8 '13 at 14:15
• Even if we set the domain as R \ {(1/2)n*pi}, the "limit" is not really a limit. It "Oscillates" by "starting" at 1/e, spending a fair amount of it's time near 1, then buzzing off to e. Rinse and repeat this process. So the limit does not exist. – Adam Rubinson Aug 8 '13 at 14:26
• Joe Oliver - The correct question would then be "What is limsup...?" – Adam Rubinson Aug 8 '13 at 14:35
• "Did answered correctly" Two characters disagree. – Did Nov 19 '13 at 15:43

$$\lim_{n\to\color{red}{\pi/2}}\left(1+\frac{1}{\tan(n)}\right)^{\tan(n)} = e$$
What is the correct question to give the answer $\mathrm e$?
$$\lim_{n\to \infty}\left(1+\tan(1/n)\right)^{1/\tan(1/n)} =\ ?$$