Asymptotic behavior of polynomialsLet

where , be a degree- polynomial in , and let be a constant. Use the definitions of the asymptotic notations to prove the following properties.

- If , then .
- If , then .
- If , then .
- If , then .
- If , then .

**1. Asymptotic Upper Bound**

To show that when , we have to show there exists constants such that for all .

The last line follows from the fact that as , for all possible values of , , i.e. all will be less than or equal to 1 for all .

So we can say that , where and .

Hence, .

**2. Asymptotic Lower Bound**

To show that when , we have to show there exists constants such that for all .

So we can say that , where and .

Hence, .

**3. Asymptotic Bound**

We have already proved that for , and . Hence, .

**4. Asymptotic Tight Upper Bound**

We can easily prove this by removing the equality from the proof of 1.

Other than that we can use the limit definition of -notation:

Hence, .

**5. Asymptotic Tight Lower Bound**

We can easily prove this by removing the equality from the proof of 2.

Other than that we can use the limit definition of -notation:

Hence, .