Limits: Uniqueness of the Limit

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 $\lim\limits_{x \to a} f(x) = L_1$ and $\lim\limits_{x \to a} f(x) = L_2$, then $L_1 = L_2$.

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

Proof Outline

1. Let $\varepsilon > 0$ be arbitrary.

2. Since $\lim\limits_{x \to a} f(x) = L_1$, there exists $\delta_1 > 0$ such that $|f(x) - L_1| < \varepsilon/2$ whenever $0 < |x - a| < \delta_1$.

3. Since $\lim\limits_{x \to a} f(x) = L_2$, there exists $\delta_2 > 0$ such that $|f(x) - L_2| < \varepsilon/2$ whenever $0 < |x - a| < \delta_2$.

4. Let $\delta = \min\{\delta_1, \delta_2\}$.

5. For $0 < |x - a| < \delta$, both inequalities above hold. Then:

   • $|L_1 - L_2| = |L_1 - f(x) + f(x) - L_2| \leq |L_1 - f(x)| + |f(x) - L_2| < \varepsilon/2 + \varepsilon/2 = \varepsilon$

6. Since $\varepsilon$ is arbitrary, $|L_1 - L_2| = 0$, so $L_1 = L_2$.

Example

• Example Calculation: $\lim\limits_{x \to 4} x^2 = 16$

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.