Problem 5 - Calculus: An Intuitive and Physical Approach

Question

Prove that if \(f(x)\) has the limit \(b\) as \(x\) approaches \(a\), then \(b\) is unique; that is, there cannot be another limit, say \(c\), in addition to \(b\).

Step-by-Step Solution

Answer

By assuming that the limits b and c are different and setting epsilon to less than half the distance between b and c, we find a contradiction because f(x) cannot be simultaneously epsilon-close to both b and c. Hence, b must equal c, proving the limit is unique.

Step 1: Assume there are two limits

Assume for contradiction that the function f(x) has two different limits as x approaches a. Let's call these limits b and c, and assume that b is not equal to c.

Step 2: Define the concept of limit formally

By definition of the limit, for every epsilon greater than 0, there exists a delta_b > 0 and delta_c > 0 such that for all x in the domain of f(x), if 0 < |x - a| < delta_b then |f(x) - b| < epsilon, and if 0 < |x - a| < delta_c then |f(x) - c| < epsilon.

Step 3: Establish contradiction using epsilon

Let epsilon be less than half the distance between b and c, so epsilon < |b - c| / 2. There must exist x values within both delta_b and delta_c intervals where f(x) is epsilon-close to both b and c, which is a contradiction since epsilon < |b - c| / 2 implies that f(x) cannot be epsilon-close to both b and c at the same time.

Key Concepts

Epsilon-Delta Definition

The epsilon-delta definition is the formal mathematical notation for defining the limit of a function as it approaches a certain point. It is a fundamental concept in calculus that captures the intuitive idea that a function gets arbitrarily close to a limit at a point.

Under the epsilon-delta definition, we say that the limit of function \(f(x)\) as \(x\) approaches \(a\) is \(L\) if for every \(\epsilon\) (epsilon), no matter how small, there exists a \(\delta\) (delta) such that for all \(x\), if \(0 < |x - a| < \delta\) then \(|f(x) - L| < \epsilon\). In simpler terms, this means that you can make \(f(x)\) as close as you want to \(L\) by choosing \(x\) sufficiently close to \(a\), but not equal to \(a\).

This definition is a rigorous way to capture the behavior of functions near a point and sets a standard framework for proving properties about limits, such as their uniqueness.

Uniqueness of Limits

In the context of limits in calculus, the uniqueness of limits states that a function can have at most one limit as it approaches a particular point. This is an important property as it guarantees that limits, when they exist, provide a definitive behavior of the function near that point.

To understand this concept, take the expression \(\lim_{x \to a} f(x) = b\). This means that as \(x\) gets very close to \(a\), the function \(f(x)\) gets very close to the value \(b\). Now, if there were another number \(c\) where \(\lim_{x \to a} f(x) = c\), this would imply that \(f(x)\) gets close to two different values at the same time, which is impossible. Hence, the limit, if it exists, must be unique.

The uniqueness of limits is critical for the predictability and stability of mathematical models. It ensures that the behavior of a function around a point is well-defined and not open to multiple interpretations.

Proof by Contradiction

Proof by contradiction is a powerful mathematical technique employed to prove a statement by showing that assuming the opposite leads to an illogical or impossible conclusion.

To use proof by contradiction, we start by assuming that the statement we want to prove is not true. In the case of proving the uniqueness of limits, we would assume that the function \(f(x)\) has two different limits as \(x\) approaches \(a\), say \(b\) and \(c\). We then show that this assumption leads to a contradiction with established facts, in this case, the epsilon-delta definition of a limit. By assuming two limits and choosing an \(\epsilon\) smaller than half the distance between the supposed limits, we reach a contradiction since \(f(x)\) cannot get close to both \(b\) and \(c\) at the same time.

This technique is highly valued in mathematics for its ability to establish certainty about statements. When properly applied, proof by contradiction can demonstrate the soundness of theory by highlighting the irrationality of any opposing proposition.