Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 9 Exercise 9.8

Solution:

### Part a

We have $\lim n^3=\infty$.

### Part b

We have $\lim(-n^3)=-\infty$.

### Part c

We have $\lim(-n)^n$ does not exists and is not equal to $\infty$ or $-\infty$ as it becomes very large but sometimes positive and sometimes negative.

### Part d

We have $\lim(1.01)^n=\infty$ by Theorem 9.7(b) and Theorem 9.10 since $\lim\left(\dfrac{1}{1.01}\right)^n=0$.

### Part e

We have $\lim (n)^n=\infty$.

If you would like to figure out why. You may want to look at Exercise 9.9.