Answer
See the limits below.
Work Step by Step
The six limits listed in Theorem 5 are
1. $\displaystyle \lim_{n\rightarrow\infty}\frac{\ln n}{n}=0.$
2. $\displaystyle \lim_{n\rightarrow\infty}\sqrt[n]{n}=1.$
3. $\displaystyle \lim_{n\rightarrow\infty}x^{1/n}=1$ for $x\gt 0.$
4. $\displaystyle \lim_{n\rightarrow\infty}x^{n}=0$ for $|x|\lt 1.$
5. $\displaystyle \lim_{n\rightarrow\infty}(1+\frac{x}{n})^{n}=e^{x}.$
6. $\displaystyle \lim_{n\rightarrow\infty}\frac{x^{n}}{n!}=0$
