limits of sequences exponential and factorial: $a_n=e^{5\cos((\pi/6)^n)}$ and $a_n=\frac{n!}{n^n}$
(a) $$ \lim_{n \to \infty} e^{5 \cos((\frac{\pi}6))^n} = e^{\lim_{n \to \infty} 5 \cos((\frac{\pi}6))^n} = e^{5 \cos(\lim_{n \to \infty}(\frac{\pi}6)^n)} = e^{5 \cdot 1} = e^5 $$
(b) $$ \frac{n!}{n^n} = \frac{1\cdot 2 \cdots n}{n\cdot n \cdot n} \leq \frac{1}{n} $$ From here $$ \lim_{n \to \infty} \frac{n!}{n^n} = 0 $$ since $$ \lim_{n \to \infty} \frac{1}{n} = 0 $$