Limits of Common Sequences

The following common sequences converge to the given limits:

1) $\lim_\limits{n\to\infty} \frac{\ln(n)}{n} = 0$

2) $\lim_\limits{n\to\infty} b^{\frac{1}{n}} = 1 \quad \text{ for } b > 0$

3) $\lim_\limits{n\to\infty} \left(1 + \frac{x}{n} \right)^n = e^x \quad \text{ (for any x)}$

4) $\lim_\limits{n\to\infty} \sqrt[n]{n} = 1$ (i.e. $(n)^{\frac{1}{n}}\to 1 \text{ as } n\to\infty$)

5) $\lim_\limits{n\to\infty} a^n = 0 \quad \text{ if } |a| < 1$

6) $\lim_\limits{n\to\infty} \frac{x^n}{n!} = 0$ (i.e. Growth of $n! > x^n$)