Proving the Limit Using Epsilon-Delta Definition: An In-depth Analysis
In this article, we delve into the epsilon-delta definition of a limit to prove the statement:
For every epsilon; 0, there exists a delta; 0 such that 0 |x - 1/3| delta; implies |1/x - 3| epsilon;.Understanding the Epsilon-Delta Definition
The epsilon-delta definition is a fundamental concept in calculus that provides a formal way to define the limit of a function. It states that for every positive value of epsilon;, there exists a positive value of delta; such that the function g(x) is within epsilon; of its limit L whenever x is within delta; of c, with x not equal to c. In mathematical terms, it can be written as:
For every epsilon; 0, there exists a delta; 0 such that 0 |x - c| delta; implies |g(x) - L| epsilon;.
Proving the Limit: Detailed Steps
To prove the limit for the expression 1/x as x approaches 1/3, we will follow these steps:
Start with the expression we need to analyze:
|1/x - 3|.
Rewrite the expression using algebraic manipulations:
|1/x - 3| |(1 - 3x) / x| 3 |(x - 1/3) / x|
.Ensure that the denominator is never zero, preventing us from bounding it. We can do this by ensuring x - 1/3 ne; 0. This can be achieved by choosing any positive quantity less than 1/3. For simplicity, choose 1/6. Therefore, we have:
x - 1/3 1/6 which implies 1/6 x 1/2. This ensures x 1/2 and thus 1/x 2.Substitute this back into our original expression to simplify it:
|1/x - 3| 3 * 2 |x - 1/3| 6 |x - 1/3|.
Taking delta;1/6 and iota;delta;epsilon;/18, we need to find a delta; such that:
0 |x - 1/3| delta; epsilon;/18 implies |1/x - 3| 18|x - 1/3| 18(epsilon;/18) epsilon;.
Conclusion
By choosing delta; min{1/6, epsilon;/18}, we can guarantee that the required condition holds. This means for any given epsilon; 0, we can find a corresponding delta; 0 such that 0 |x - 1/3| delta; implies |1/x - 3| epsilon;.
Therefore, the limit as x approaches 1/3 of 1/x is 3.
Key Takeaways
1. **Epsilon-Delta Definition**: This rigorous formalism is used to define the limit of a function accurately. 2. **Algebraic Manipulation**: Simplifying the expression is crucial to prove the limit. 3. **Non-Zero Denominators**: Ensuring the denominator is never zero is vital to avoid undefined values.