What is the Least Natural Number n for Which √(x² + x³ + 3) is O(xⁿ)?

[hain]

I'm having some trouble with this discrete question:

Find the least natural number n such that
√(x² + x³ + 3) is O(xⁿ).

With the value of n that you have found, is it true that
xⁿ is O( √(x² + x³ + 3) )?


Can anyone help?

 

[HallsofIvy]

Well, starting with the definition of "O" would be a good idea. What is it?

 

[hain]

http://en.wikipedia.org/wiki/Big-O_notation

I'm able to determine n in simpler cases, but I have no idea how to approach the root of this polynomial.

 

[Hurkyl]

These are easy: they're close enough to polynomials for the purposes asymptotic analysis. How would you do it if it was a polynomial?

 

[hain]

Err, typing this stuff out is difficult. Here's an example with polynomials: http://en.wikipedia.org/wiki/Big-O_notation#Example . It's images so it's easy to read.

 

[HallsofIvy]

Why would you want to find a root? What is
\frac{\sqrt{x^2+x^3+ 3}}{x^n}
?