<s>[INST] Evaluate $\lim\limits_{n\to \infty} \left( \cos(1/n)-\sin(1/n) \right) ^n $? How to calculate 
$\lim\limits_{n\to \infty} \left( \cos(1/n)-\sin(1/n) \right) ^n $?
Since $\lim\limits_{n\to \infty} \frac {\cos(1/n)-\sin(1/n) }{1-1/n} = 1  $, I guess that the limit above is $\frac{1}{e}$, but since the form $(\to 1)^{\to \infty}$  is indeterminate, I don't know how to prove it formally.
Thanks!
 [/INST] You may consider 


*

*$\left(\frac{\cos x - \sin x}{1-x}\right)^{\frac{1}{x}}$ for $x\to 0$ and use

*$\lim_{t\to 0} (1+t)^{\frac{1}{t}}= e$
\begin{eqnarray*} \left(\frac{\cos x - \sin x}{1-x}\right)^{\frac{1}{x}}
& = & \left( \underbrace{\left(1 + \frac{\cos x - \sin x -1 +x}{1-x}\right)^{\frac{1-x}{\cos x - \sin x -1 +x}}}_{\stackrel{x\to 0}{\longrightarrow}e}\right)^{\underbrace{\frac{\cos x - \sin x -1 +x}{x-x^2}}_{\stackrel{L'Hop}{\sim}\frac{-\sin x - \cos x +1}{1-2x}\stackrel{x\to 0}{\longrightarrow}0}}\\
& \stackrel{x\to 0}{\longrightarrow} & e^0 = 1
\end{eqnarray*}
</s>