concerning big-oh notation and proving or disproving

Question

So I'm completely lost on big-oh notation.

In my assignment I am supposed to prove or disprove the following using the formal definition.

3n³ - 7n² + 100n - 36 is in O(n³)

and

n²/log(n) + 3n is in O(n²)

Can someone help me with these and tell me how to go about proving or disproving.

Answer 1

The definition of Big-O is

f(n) ∈ O(g(n)) ⇔ lim supn → ∞ | f(n)/g(n) | < ∞

Since your f(n) and g(n) are polynomials lim sup and lim are the same. So you have to proof:

1. limn → ∞ |(3n³ - 7n² + 100n - 36) / n³| < ∞

   limn → ∞ |(3n³ - 7n² + 100n - 36) / n³| 
    = limn → ∞ |n³(3 - 7/n + 100/n² - 36/n³) / n³|
    = limn → ∞ |3 - 7/n + 100/n² - 36/n³|
    = 3 < ∞
   

2. lim supn → ∞ |(n²/log(n) + 3n) / n²| < ∞

   lim supn → ∞ |(n²/log(n) + 3n) / n²| 
    = lim supn → ∞ |n²(1/log(n) + 3/n) / n²|
    = lim supn → ∞ |1/log(n) + 3/n|
    = 0 < ∞
   

   here you can also see that n²/log(n) + 3n is also in o(n²).