MHB Optimizing B for Inequality with Floor Function

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SUMMARY

The discussion focuses on finding a positive value of B that satisfies the inequality condition involving the floor function and a given positive A. Specifically, for all x greater than B, the expression |x/(x - ⌊x²⌋| must be less than A. The participants clarify that the limit concept should not be employed, and they derive that the expression can be simplified to |1/x| * |1/(1/x - ⌊x²⌋/x²)|. This simplification is crucial for determining the appropriate bounds for B in relation to A.

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  • Understanding of floor functions and their properties
  • Familiarity with inequalities and their manipulation
  • Basic knowledge of limits and asymptotic behavior
  • Proficiency in algebraic simplification techniques
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  • Research the properties of floor functions in mathematical analysis
  • Study inequalities and their applications in optimization problems
  • Explore algebraic techniques for simplifying complex expressions
  • Learn about asymptotic analysis and its relevance in bounding functions
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Mathematicians, students in advanced calculus or analysis courses, and anyone interested in optimization problems involving inequalities and floor functions.

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given $A>0$ find a $B>0$ such that:

For all $x>0$ and $x>B$ Then $|\frac{x}{x-\lfloor x^2\rfloor}|<A$
Do not use the concept of the limit
 
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hint:[sp] $|\frac{x}{x-\lfloor x^2\rfloor}|$=$|\frac{1}{x}|.|\frac{1}{\frac{1}{x}-\frac{\lfloor x^2\rfloor}{x^2}}|$[/sp]
 

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