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

In summary, the precise definition of a limit uses |f(x)-L| < ε instead of ≤ ε because it is more natural and intuitive. With the delta-epsilon definition, a range of values for x is defined within a certain window (δ), which also defines a window around a range of y values (ε). If the definition were changed to ≤ ε, the window would be included as part of the range, making it less intuitive. This discussion took place last year.
  • #1
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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  • #2
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.
 
  • #3
This thread is from last year. Please be careful to check the date when posting.
 

1. What is the definition of a limit?

The definition of a limit is a mathematical concept that describes the behavior of a function as the input values approach a certain point. It is used to determine the value that a function approaches as its input values get closer and closer to a specific point.

2. Why is it important to choose a value for |f(x)-L| that is less than ε?

Choosing a value for |f(x)-L| that is less than ε is important because it ensures that the function is approaching a specific point as closely as possible. By choosing a value for ε, we can control how close the function is getting to the limit, and therefore, have a better understanding of its behavior.

3. What happens if |f(x)-L| is equal to ε?

If |f(x)-L| is equal to ε, it means that the function is approaching the limit exactly. This can be thought of as the function "hitting" the limit point. In this case, we can say that the function is continuous at that point.

4. Can the value of ε be negative?

No, the value of ε cannot be negative. It represents a distance or a difference between the function and the limit, so it must be a positive value. A negative value would not make sense in this context.

5. How does the definition of a limit differ from the value of a function at a specific point?

The definition of a limit focuses on the behavior of a function as the input values get closer to a specific point, while the value of a function at a specific point is simply the value of the function at that point. The concept of a limit allows us to understand the overall behavior of the function, rather than just its value at one point.

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