Definition of Limit: Why Choose |f(x)-L| < ε?

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The discussion centers on the delta-epsilon definition of limits in calculus, specifically why the condition |f(x)-L| < ε is preferred over |f(x)-L| ≤ ε. Participants agree that using < ε creates a more intuitive understanding of limits by defining a strict range of values for f(x) without including the boundary. This approach emphasizes the concept of approaching a limit rather than reaching it, which aligns with the foundational principles of calculus.

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doubleaxel195
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This isn't really a homework problem, but I was wondering why in the precise definition of a limit do we choose to make [tex]|f(x)-L| < ε[/tex] and not less than or equal to ε? I was just wondering. I asked my professor, he said he'd think about it, but he never got back to me.
 
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Technically you could define it as ≤ ε, but it would be less natural. With the delta-epsilon definition, you define a range of values for x within a certain window (δ). For the given function, this defines a window around a range of y values (ε). If you change it to ≤ ε, you include the window as part of the range. It is more intuitive if the range of values is within a given window or < ε.
 
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