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