Proving the Limit of x → 0 Using the δ-ε Definition

One of the fundamental concepts in calculus is the limit of a function. The limit of a variable x approaching a value a is a precise way to describe the behavior of f(x) as x gets arbitrarily close to a. Proving such a limit rigorously involves the δ-ε definition. In this article, we will explore how to prove that the limit of x as x approaches 0 is 0, using the δ-ε definition.

Understanding the δ-ε Definition

The δ-ε definition of a limit is a formal definition that allows us to precisely state what it means for a function's value to approach a certain limit as the input approaches a specific value. It is stated as follows:

Let f be a function defined on an open interval containing a, except possibly at a itself. Then we say that the limit of f(x) as x approaches a is L if for every ε > 0, there exists a δ > 0 such that for all x, 0

In simpler terms, for any small positive number ε, we can find a corresponding positive number δ, such that whenever x is within δ of a but not equal to a, the value of f(x) is within ε of the limit L.

Applying the δ-ε Definition to the Limit of x as x Approaches 0

Let's apply the δ-ε definition to prove that the limit of x as x approaches 0 is 0.

Step 1: Understand the Problem

We need to show that for every ε > 0, there exists a δ > 0 such that for all x, 0

Step 2: Choose δ

Let ε > 0 be given. We will choose δ ε. This is our candidate for the value of δ that will satisfy the δ-ε condition.

Step 3: Prove the Condition

Suppose that 0

Conclusion

We have shown that for every ε > 0, we can choose δ ε such that for all x, 0

limx→0 x 0

Key Points to Remember

The δ-ε definition is a rigorous way to define limits in calculus. For the limit of x as x approaches 0 to be 0, we can always find a δ such that |x| Choosing δ ε is a common strategy to prove the δ-ε condition for simple functions like the one in this example.

Related Topics and Further Reading

Understanding the δ-ε definition is crucial for mastering more advanced topics in calculus, such as continuity and differentiability.

Further Reading

MIT OpenCourseWare: Introduction to Calculus - limits and continuity Coursera: Calculus: Single Variable - limits and derivatives Paul's Online Math Notes: Limits and Continuity

If you want to delve deeper into this topic, I recommend checking out the recommended resources. There are also many online tutorials and interactive exercises available to help you practice and understand the δ-ε definition of limits better.