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

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Homework Help Overview

The discussion revolves around determining the least natural number n for which the expression √(x² + x³ + 3) is classified as O(xⁿ). Participants are exploring the implications of Big-O notation in the context of asymptotic analysis.

Discussion Character

  • Conceptual clarification, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants are questioning the definition of Big-O notation and its application to the given expression. There is an exploration of how to analyze the root of the polynomial and its relation to polynomial behavior in asymptotic terms.

Discussion Status

The discussion is ongoing, with participants providing references to definitions and examples related to Big-O notation. Some are attempting to relate the problem to simpler polynomial cases, while others are questioning the necessity of finding a root in this context.

Contextual Notes

There is a mention of the difficulty in typing out mathematical expressions, indicating a potential barrier to clear communication of ideas. Additionally, the original poster seeks assistance in understanding the problem setup and implications.

hain
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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?
 
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Well, starting with the definition of "O" would be a good idea. What is it?
 
These are easy: they're close enough to polynomials for the purposes asymptotic analysis. How would you do it if it was a polynomial?
 
Why would you want to find a root? What is
[tex]\frac{\sqrt{x^2+x^3+ 3}}{x^n}[/tex]
?
 

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