Ch. 1 - Limits

3.4 Proposição: Unicidade do Limite (Uniqueness of the Limit)

The uniqueness of the limit is a fundamental property in calculus, ensuring that a function cannot approach two different values as the input approaches a particular point. This section presents the formal statement and proof of this property.

• Statement: If and , then .

• Key Concept: The limit of a function at a point, if it exists, is unique.

Proof Outline

1. Let be arbitrary.

2. Since , there exists such that whenever .

3. Since , there exists such that whenever .

4. Let .

5. For , both inequalities above hold. Then:

6. Since is arbitrary, , so .

Example

• Example Calculation:

Additional info: The proof uses the triangle inequality and the definition of the limit to show that two supposed limits must be equal, reinforcing the consistency of calculus.