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

[doubleaxel195]

This isn't really a homework problem, but I was wondering why in the precise definition of a limit do we choose to make $$|f(x)-L| < ε$$ 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.

 

[math man]

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.

 

[micromass]

This thread is from last year. Please be careful to check the date when posting.