Proving the Limit of a Function Using ε-δ Definition: An Example

In the field of mathematical analysis, the ε-δ definition is a fundamental concept used to formally define limits. This article provides a step-by-step proof of the limit of the function limx-0 1/sqrt(1 x^2) 1/2 using the ε-δ definition.

Understanding the ε-δ Definition

The ε-δ definition states that the limit of a function f(x) as x approaches a is L if for every positive number ε (epsilon), there exists a positive number δ (delta) such that whenever 0 |x - a| δ, the inequality |f(x) - L| ε holds. This definition requires us to find a suitable δ for any given ε to ensure the desired condition is met.

Applying the ε-δ Definition to Prove the Limit

Let's prove that limx-0 1/sqrt(1 x^2) 1/2. According to the ε-δ definition, we need to find δ such that for every positive ε, the following inequality holds: |1/sqrt(1 x^2) - 1/2| ε for every x in the interval -(δ x δ).

Step 1: Simplify the Expression

First, we simplify the expression inside the absolute value: | 1/sqrt(1 x^2) - 1/2 |

We can rewrite this as: | 1/2 - 1/sqrt(1 x^2) |

Combine the fractions over a common denominator:

Step 2: Isolate the Square Root Term

Next, we isolate the square root term in the numerator: | sqrt(1 x^2) - 1 | / (2 * sqrt(1 x^2))

Multiplying both sides by the conjugate (sqrt(1 x^2) 1), we get: | (sqrt(1 x^2) - 1)(sqrt(1 x^2) 1) | / (2 * sqrt(1 x^2) * (sqrt(1 x^2) 1))

This simplifies to: | (1 x^2 - 1) / (2 * sqrt(1 x^2) * (sqrt(1 x^2) 1)) |

Further simplifying, we get: | x^2 / (2 * sqrt(1 x^2) * (sqrt(1 x^2) 1)) |

Step 3: Choose an Appropriate δ

For the inequality | x^2 / (2 * sqrt(1 x^2) * (sqrt(1 x^2) 1)) | ε to hold, we need: x^2 / (2 * sqrt(1 x^2) * (sqrt(1 x^2) 1)) ε

Rearranging this, we get: x^2 2 * ε * sqrt(1 x^2) * (sqrt(1 x^2) 1)

For sufficiently small δ, we can approximate the denominator as follows:

This simplifies to: x^2 2 * ε * (1 x^2 x^2/2)

Further simplifying, we get: x^2 2 * ε * (1 3x^2/2)

For the smallest δ, we can choose: x^2 8 * ε

Hence, we can choose δ as: δ sqrt(8ε)

Conclusion

By selecting δ sqrt(8ε), we have demonstrated that for any positive ε, there exists a positive δ such that the inequality |1/sqrt(1 x^2) - 1/2| ε holds whenever 0 |x| δ. This proves the limit using the ε-δ definition.

Key Takeaways

The ε-δ definition is a rigorous approach to proving limits in mathematical analysis. To prove a limit, we need to find a δ such that for any given ε, the inequality holds. Manipulating algebraic expressions and approximations play a crucial role in determining the correct δ.

Related Keywords

ε-δ Definition, Limit Proof, Mathematical Analysis