Big Oh Notation Explained: What Does x = x_0 + \mathcal{O}(y) Mean?

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SUMMARY

The discussion clarifies the meaning of the expression x = x_0 + \mathcal{O}(y) in the context of Big Oh notation. It emphasizes the necessity of including a limit, specifically "as x goes to...", to provide meaningful context to the expression. The example given, \cos x = 1 + O(x^2) as x \to 0, illustrates that the term \frac{\cos x - 1}{x^2} is bounded in a neighborhood around 0, demonstrating the practical application of Big Oh notation in analyzing function behavior near specific points.

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  • Understanding of Big Oh notation
  • Familiarity with limits in calculus
  • Basic knowledge of trigonometric functions
  • Ability to analyze function behavior near specific points
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  • Study the formal definition of Big Oh notation
  • Learn about limits and their applications in calculus
  • Explore Taylor series expansions for trigonometric functions
  • Investigate boundedness and continuity in mathematical analysis
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Mathematicians, computer scientists, and students studying algorithms or calculus who seek to deepen their understanding of function behavior and complexity analysis using Big Oh notation.

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Can someone please explain what it means to say something like
[tex] x = x_0 + \mathcal{O}(y)[/tex]
?
 
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Nothing.

You must include in your expression a "as x goes to.."

Without that, it is meaningless.
 
[itex]\cos x = 1 + O(x^2)[/itex] as [itex]x \to 0[/itex] means:
[tex]\frac{\cos x - 1}{x^2}[/tex]
is bounded in some neighborhood of [itex]0[/itex] .
 

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