Solving Simple Homework Problems: An Exercise in O(x^4) Analysis

[TheLegace]

Homework Statement

Hi, I have been having huge problems with dealing with these kinds of problems, I would appreciate atleast some guidance in dealing with these sorts of problems, I think the major problem is just how I learned to solve them, I have been looking for resources on the net, but it just gets to complicated, and I need someone to help me start with simple stuff first.

A very simple problem I am starting with will be this one:

Is x^4 + 9x^3 + 4x + 7 O(x^4) ?

Homework Equations

Well obviously start with:

|x^4 + 9x^3 + 4x + 7| ≤ C|x^4|

The Attempt at a Solution

Now if I try working this out

|x^4 + 9x^3 + 4x + 7| ≤ C|x^4| for all x > k
x^4 + 9x^3 + 4x + 7 ≤ 1x^4 + 9x^4 4x^4 7x^4 for all x > 1
f(x) ≤ 21x^4 for all x > 1
so for C=21 and k=1 f(x) = O(x^4).

Now the solution in the textbook says the constants are C=4, k=9.
Am I wrong or do the constants matter, I know there are infinite amount of constants if the statement is true, but why choose those ones anyway then?

Any help would be appreciated.
Thank You.

 

[Billy Bob]

I like your way. Simple and direct. The actual constants don't matter.