MHB Optimizing B for Inequality with Floor Function

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To optimize B for inequality with the floor function, the goal is to find a B>0 such that for all x>0 and x>B, the expression |(x/(x-⌊x²⌋)| is less than A. The discussion emphasizes avoiding the limit concept and instead focuses on manipulating the expression to show that |(x/(x-⌊x²⌋)| can be rewritten as |(1/x)| * |(1/(1/x - ⌊x²⌋/x²))|. The participants explore how to derive conditions on B that ensure the inequality holds true for all x greater than B. The conversation highlights the mathematical intricacies involved in bounding the expression effectively. Ultimately, the focus remains on establishing the right value for B to satisfy the given condition.
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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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