Proving the Limit Does Not Exist Using the Epsilon-Delta Definition

In this article, we delve into the mathematical proof that the limit (lim_{x to x_0} f(x)) does not exist, utilizing the epsilon-delta definition. This important concept in real analysis is crucial for understanding the nature of functions and their behavior around certain points. We will explore the key ideas through a step-by-step proof, enhancing your understanding of the epsilon-delta method.

Key Definitions and Concepts

First, let's establish some basic definitions and notation:

(epsilon): A small positive real number used to specify the closeness of the function values to the limit. (delta): A positive real number that defines the size of the neighborhood around (x_0). (f(x)): The function whose limit we are considering. (x_0): The point around which we are examining the limit. (L): The hypothetical limit value.

Proof via Contradiction

Let's assume that the limit (lim_{x to x_0} f(x) L) exists. According to the epsilon-delta definition of a limit, for any (epsilon > 0), there exists a (delta > 0) such that for all (x) in the domain of (f), if (0 , then (|f(x) - L| .

Given this, let's set (epsilon frac{1}{2}beta) where (beta) is a positive real number. By the definition of the limit, there exists a (delta > 0) such that for all (x_1, x_2) in the domain of (f) and satisfying (0 and (0 , we have:

(|f(x_1) - L| geq frac{1}{2}beta) (|f(x_2) - L| geq frac{1}{2}beta)

From the triangle inequality, we know:

(|f(x_1) - f(x_2)| leq |f(x_1) - L| |L - f(x_2)|)

Thus, we have:

(|f(x_1) - f(x_2)| leq frac{1}{2}beta frac{1}{2}beta beta)

However, since not both (|f(x_1) - L|) and (|f(x_2) - L|) can be less than (beta/2), at least one of them must be equal to or greater than (beta/2). This contradicts the requirement that (beta . Therefore, the limit does not exist.

Another Approach: Direct Contradiction

Suppose, for the sake of contradiction, that the limit does exist and equals (b). Let (delta frac{beta}{2}). By the definition of the limit, there exists a neighborhood (V) of (x_0) such that for all (x) in (V), we have:

(|f(x) - b| frac{beta}{2})

Now, take (x_1, x_2 in V). We have:

(|f(x_1) - f(x_2)| |f(x_1) - b b - f(x_2)| leq |f(x_1) - b| |b - f(x_2)| frac{beta}{2} frac{beta}{2} beta)

By the triangle inequality, we know:

(|f(x_1) - f(x_2)| leq beta)

However, for the limit to exist, the function values (f(x_1)) and (f(x_2)) must be arbitrarily close around the limit value (b). Since their difference can be at most (beta), which is a fixed positive number, this contradicts the strict conditions of the epsilon-delta definition. Therefore, the limit does not exist.