Proving the Limit of a Function Using the Epsilon-N Definition
In the realm of mathematical analysis, the precise definition of a limit, often referred to as the epsilon-N (or epsilon-delta in some contexts) definition, is a fundamental tool. This article explores how to prove a limit using this definition, specifically in the case where x approaches infinity. We will demonstrate the process through a detailed example involving the function f(x) (sqrt{x} - 1) / (x^2 - 1) as x goes to infinity.
Introduction to the Epsilon-N Definition
The epsilon-N definition of a limit is a rigorous statement that addresses the behavior of a function as the input, x, grows without bound. It states that a function f(x) approaches a limit L as x goes to infinity if, for every positive epsilon, there exists a positive integer N such that for all x ≥ N, the value of f(x) satisfies the inequality |f(x) - L| epsilon.
Function and Initial Setup
Given the function f(x) (sqrt{x} - 1) / (x^2 - 1), we aim to prove that as x approaches infinity, f(x) approaches 0. This means we need to show that for every positive epsilon, there exists a positive integer N such that for all x ≥ N, we have:
|(sqrt{x} - 1) / (x^2 - 1) - 0| epsilon
Algebraic Manipulation
Let's start by simplifying the expression |f(x) - 0| for x ≥ 1.
[ left| frac{sqrt{x} - 1}{x^2 - 1} - 0 right| left| frac{sqrt{x} - 1}{x^2 - 1} right| ]
We can further simplify this expression using algebraic manipulation:
[left| frac{sqrt{x} - 1}{x^2 - 1} right| frac{|sqrt{x} - 1|}{|x^2 - 1|} cdot frac{|sqrt{x} 1|}{|sqrt{x} 1|}]Note that the term x^2 - 1 can be factored as (x - 1)(x 1). Thus:
[left| frac{sqrt{x} - 1}{x^2 - 1} right| frac{|sqrt{x} - 1|}{|(x - 1)(x 1)|} cdot (sqrt{x} 1)]Given that x ≥ 1, we can simplify this further:
[left| frac{sqrt{x} - 1}{x^2 - 1} right| frac{|sqrt{x} - 1|}{|(x - 1)(x 1)|} cdot (sqrt{x} 1) leq frac{2}{x^{3/2}} ]Choosing N Based on Epsilon
To ensure that the inequality |f(x) - 0| epsilon holds, we need to find a suitable choice for N in terms of epsilon. From the simplified expression, we see that as x increases, the value of 2 / x^(3/2) decreases, approaching 0.
Specifically, we need:
[frac{2}{x^{3/2}} epsilon]Rearranging this inequality, we derive:
[frac{1}{x^{3/2}} frac{epsilon}{2}]Taking the reciprocal of both sides (and considering that the function is positive), we get:
[x^{3/2} frac{2}{epsilon}]Raising both sides to the 2/3 power:
[x left( frac{2}{epsilon} right)^{2/3}]Therefore, we can choose N to be:
[N leftlceil left( frac{2}{epsilon} right)^{2/3} rightrceil]Here, the ceiling function ensures that N is a positive integer.
The Proof
Given any positive epsilon, we have established that for all x ≥ N, where N (2/epsilon)^(2/3), the following holds:
[left| frac{sqrt{x} - 1}{x^2 - 1} right| epsilon]This completes the proof that as x approaches infinity, the function f(x) (sqrt{x} - 1) / (x^2 - 1) approaches 0.
Conclusion
In this article, we demonstrated a step-by-step approach to proving a limit using the epsilon-N definition. The key was to recognize the dominant terms in the function and use algebraic manipulation to simplify the expression. By carefully choosing N based on epsilon, we were able to ensure that the function's value remains within the desired tolerance. Understanding and applying the epsilon-N definition is crucial for grasping deeper concepts in mathematical analysis.