最紧密的渐近增长率是多少

Question

I have solved all of them however i have been told there are some mistakes, can somebody please help me

n^4 - 10^3 n^3 + n^2 + 4n + 10^6 = O(n^4)

10^5 n^3 + 10^n = O(10^n)

10 n^2 + n log n + 30 √n = O(n^2)

25^n = O(1)

n^2+ n log n + 7 n = O(n^2)

(n^3 + 10) (n log n+ 1) / 3 = O(n^4 log n)

20 n^10 + 4^n = O(4^n)

n^2 log n^3 + 10 n^2 = O(n^2 log n)

10^20 = O(1)

n^2 log (6^2)n = O(n^2 log n)

n log(2n) = O(n log n)

30 n + 100 n log n + 10 = O(n log n)

(n+√n) log n^3 = O(n+√n log n)

n (n + 1) + log log n = O(n^2)

4n log 5^(n+1) = O(n log 5^n)

3^(n+4) = O(3^n)

n^2 log n^2 + 100 n^3 = O(n^3)

(n log n) / (n + 10) = O(n^2 log n)

5n + 8 n log(n) + 10n^2 = O(n^2)

2n^3 + 2n^4 + 2^n + n^10 = O(2^n)

Answer 1

Hints:

1. if you have n on the left, you should have it on the right
2. there should not be any + operations on the right
3. log(x^y) can be simplified

Answer 2

Most of your answers look correct, but you have 25^n = O(1) which looks wrong (unless it's 0.25^n), and also you have (n log n) / (n + 10) = O(n^2 log n) which does not look like the tightest possible bound (I'm assuming you want the tightest possible upper bound function). Also you should never have to add functions in your big-O, unless your original function is taking the sum or max of two functions or something and the two functions have cris-crossing different growth rates at different values of n as n goes to infinity. And that very rarely happens.