Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
General question on function growth
#1
I've been reading this explanation.

Now take functions
f(x) = x^x
g(x) = (x + 1)^(x + 1)

According to the definition of "little-oh", I'd conclude that f(x) = o(g(x)).

Am I right?
Reply
#2
(03/08/2011, 07:37 AM)dyitto Wrote: I've been reading this explanation.

Now take functions
f(x) = x^x
g(x) = (x + 1)^(x + 1)

According to the definition of "little-oh", I'd conclude that f(x) = o(g(x)).

Am I right?

Yes, because f is not : if you would chose any constant C>0 (> 0 is essential though omitted in that text, better look at wikipedia), then you always find
x^x < C (x+1)^(x+1) for large enough x
because
x^x / (x+1)^(x+1) < (x+1)^x / (x+1)^(x+1) = 1/(x+1) < C
Reply
#3
Intuitively I would say that the above functions f and g have about the same growth rate, since f simply stays one step behind g.
A function with a REAL different growth rate would be:

h(x) = x^(x^x)

So if I wanted to look into the relative growth of hyperoperational functions, then these Bachmann–Landau notation apparently wouldn't be of much use in this context.
Reply


Possibly Related Threads...
Thread Author Replies Views Last Post
  Composition, bullet notation and the general role of categories MphLee 8 2,799 05/19/2021, 12:25 AM
Last Post: MphLee
  Math.Stackexchange.com question on extending tetration Daniel 3 1,336 03/31/2021, 12:28 AM
Last Post: JmsNxn
  A Holomorphic Function Asymptotic to Tetration JmsNxn 2 1,097 03/24/2021, 09:58 PM
Last Post: JmsNxn
  Kneser method question tommy1729 9 9,729 02/11/2020, 01:26 AM
Last Post: sheldonison
  New mathematical object - hyperanalytic function arybnikov 4 6,115 01/02/2020, 01:38 AM
Last Post: arybnikov
  Is there a function space for tetration? Chenjesu 0 2,179 06/23/2019, 08:24 PM
Last Post: Chenjesu
  A Notation Question (raising the highest value in pow-tower to a different power) Micah 8 11,029 02/18/2019, 10:34 PM
Last Post: Micah
  Degamma function Xorter 0 2,593 10/22/2018, 11:29 AM
Last Post: Xorter
  Math overflow question on fractional exponential iterations sheldonison 4 9,133 04/01/2018, 03:09 AM
Last Post: JmsNxn
  Between exp^[h] and elementary growth tommy1729 0 2,632 09/04/2017, 11:12 PM
Last Post: tommy1729



Users browsing this thread: 1 Guest(s)