# Some Problems

Some of these exercises are mathematically demanding. I suggest that to those who find this difficult, that rather than becoming discouraged, skip over the questions straight to the further reading section. These exercises are somewhat more interesting for those who are interested in mathematical analysis.

1. Prove that is .
2. Prove that is (Hint: use L'Hôpital's Rule).
3. Prove that is (Hint: Use L'Hôpital's Rule).
4. Prove if and , then (Hint: Consider the definition of big-O and the definition of limits!)
5. Prove if and , then . (Harder).
6. Prove if and , then .
7. Prove that if and , then there exists a such that .
(Hint: Use the following result: Let . Suppose there exists an such that implies . Let such that as , . Then there exists an such that for all implies . Try to prove this proposition, it's fun!)
8. Prove that in general if and implies is not true.

Menaka Lashitha Bandara 2005-04-18