Asymptotic behavior of polynomials

Let

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

  1. If , then .
  2. If , then .
  3. If , then .
  4. If , then .
  5. 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, .

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