Proving a Limit is Incorrect Using the Epsilon-Delta Definition

In calculus, the epsilon-delta definition of a limit is a fundamental concept used to rigorously define the behavior of a function as its input approaches a certain value. This article will explore how to use the epsilon-delta definition to prove that the limit of a particular function is incorrect. We will specifically demonstrate that the limit of x as x approaches 0 is not equal to 1.

What is the Epsilon-Delta Definition of a Limit?

According to the epsilon-delta definition, for a limit to exist, the following must be true:

The limit of a function f(x) as x approaches a value c is L if for every epsilon; > 0, there exists a delta; > 0 such that whenever 0 |x - c| delta;, it follows that |f(x) - L| epsilon;.

Proving the Limit of x as x Approaches 0 is Not Equal to 1

Let's consider the function f(x) x and show that the limit of f(x) as x approaches 0 is not equal to 1. We will approach this by contradiction.

Assumption and Contradiction

Assume, for contradiction, that:

limx → 0 x 1

Choose an arbitrary epsilon; 0. For our proof, we will set epsilon; 1/2. According to our assumption, there must exist a delta; 0 such that whenever 0 |x - 0| delta;, it follows that |x - 1| 1/2.

Analyzing the Inequality

Consider the inequality:

|x - 1| 1/2

This can be rewritten as:

-1/2 x - 1 1/2

Adding 1 to all parts of the inequality:

1/2 x 3/2

Considering the Range of x

If x is to satisfy 0 |x - 0| delta;, then x must be close to 0, specifically -delta; x delta;. For this problem, if we select delta; 1/2, then x will be in the interval -1/2 x 1/2.

Reaching a Contradiction

Now, let's consider delta; 1/2. For values of x in the interval -1/2 x 1/2, there will be some x that do not satisfy the inequality 1/2 x 3/2. For example, any x in the interval 0 x 1/2 will not satisfy 1/2 x 3/2. This contradiction shows that:

The assumption that limx → 0 x 1 is false.

Therefore, we conclude:

limx → 0 x ≠ 1.

Conclusion

This demonstrates that the limit of x as x approaches 0 is indeed not equal to 1. The epsilon-delta definition provides a rigorous framework to prove the correctness or incorrectness of a given limit statement.

Further Reading

Introduction to Limits and Their Properties Exploring More Complex Limits Understanding the Epsilon-Delta Definition with Examples