Using the Epsilon-Delta Definition to Prove Limits: A Mathematical Insight

In mathematical analysis, the epsilon-delta definition of a limit is a fundamental concept. This definition provides a rigorous framework to prove that a function approaches a certain value as the input approaches a specific point. This article will illustrate how to use the epsilon-delta definition to show that (lim_{x to 1} frac{x^2 - 1}{x - 1} 3). This proof is an essential tool for understanding the behavior of functions near certain points and is crucial in various fields of mathematics and its applications.

The Epsilon-Delta Proof Process

We aim to prove the following statement using the epsilon-delta definition:

(forall epsilon > 0 , exists delta > 0 , text{s.t} , |x - 1| delta implies left| frac{x^2 - 1}{x - 1} - 3 right| epsilon)

Step-by-Step Proof

Let's start by manipulating the expression inside the absolute value to make it resemble the form of (epsilon) and (delta).

First, rewrite the expression:

(frac{x^2 - 1}{x - 1} frac{(x 1)(x - 1)}{x - 1})

For (x eq 1), this simplifies to:

(x 1)

Therefore, we need to show:

(lim_{x to 1} (x 1) 3)

Simplifying the Expression

Now, we need to manipulate the expression to get it in terms of (delta) and (epsilon) as follows:

(left| frac{x^2 - 1}{x - 1} - 3 right| left| x 1 - 3 right| |x - 2|)

We want:

(|x - 2| epsilon)

This implies:

(|x - 2| epsilon implies |x - 1 - 1| epsilon implies |x - 1| 1 epsilon implies |x - 1| epsilon - 1)

Choosing (delta)

From the above inequality, we choose:

(delta text{min}{1, epsilon - 1})

This ensures that (|x - 1| delta) implies (|x - 2| epsilon).

Conclusion

Thus, for any given (epsilon > 0), we can find a (delta > 0) (specifically, (delta text{min}{1, epsilon - 1})) such that:

(|x - 1| delta implies left| frac{x^2 - 1}{x - 1} - 3 right| epsilon)

This completes the proof that (lim_{x to 1} frac{x^2 - 1}{x - 1} 3).

Mathematically, we write:

(lim_{x to 1} frac{x^2 - 1}{x - 1} 3)

This proof showcases the power of the epsilon-delta definition in rigorously proving the continuity of functions at specific points.