A homework about growth rate of function

Question

Please order the function belows by growth rate

n ^ 1.5
n ^ 0.5 + log n
n log ^ 2 n
n log ( n ^ 2 )
n log log n
n ^ 2 + log n
n log n
n

ps: Ordering by growth rate means, as n gets larger and larger, which function will eventually be higher in value than the others.

ps2. I have ordered most of the functions: n , n log log n, n log n, n log^2 n, n log ( n ^ 2 ), n ^ 1.5

I just do not know how to order: n ^ 2 + log n, n ^ 0.5 + log n, these 2 values

Can anyone help me? Thank you

Answer 1

You can figure this out fairly easily by graphing the functions and seeing which ones get larger (find a graphing calculator, check out Maxima, or try graphing the functions on Wolfram Alpha). Or, or course, you just pick some large value of n and compare the various functions, but graphs can give a bit of a better picture.

Answer 2

The key to the answer you seek is that when you sum two functions, their combined "growth rate" is going to be exactly that of the one with the higher growth rate of the two. So, you now know the growth rates of these two functions, since you appear (from knowing the correct ordering of all the others) to know the proper ordering of the growth rates that are in play here.