Problem 88 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(\lim _{x \rightarrow 0} f(x)=3\), then \(f(0)=3\).

Step-by-Step Solution

Answer

The statement is false. The limit \(\lim_{x\rightarrow 0} f(x)=3\) does not necessarily imply that \(f(0)=3\). A counterexample is the function \(g(x) = \begin{cases} \frac{\sin x}{x} & \text{if} \ x \neq 0 \\ c & \text{if} \ x = 0 \end{cases}\), where \(c\) is any real number. The limit \(\lim_{x\rightarrow 0} g(x) = 1\), but the value of the function at x=0 depends on the chosen value for \(c\), which can be different from the limit.

Step 1: Basic Limit Definition

A limit is defined as the value a function approaches as the input variable approaches a specific value. In this case, the limit of f(x) as x approaches 0 is 3, which means that f(x) gets arbitrarily close to 3 as x gets arbitrarily close to 0. However, the limit itself does not say anything about the value of the function at the specific point (i.e., x=0 in this case).

Step 2: Determine Truth Value

Since the limit definition does not specify the value of the function at the point itself, the given statement: "If \(\lim_{x\rightarrow 0} f(x) = 3\), then \(f(0)=3\)" is not necessarily true.

Step 3: Counterexample

Consider the function \(f(x) = \frac{\sin x}{x}\), and a modified function defined as: \[ g(x) = \begin{cases} \frac{\sin x}{x} & \text{if} \ x \neq 0 \\ c & \text{if} \ x = 0 \end{cases} \] where \(c\) is any real number. For the function \(g(x)\), we have the limit \(\lim_{x\rightarrow 0} g(x) = \lim_{x\rightarrow 0} \frac{\sin x}{x} = 1\). However, by selecting different values for \(c\), we can see that for each case, the limit is still 1 while the value of the function at x=0 varies. For example, if we choose \(c = 3\), then \(g(0) = 3\) which satisfies the statement. But if we choose a different value, say \(c = 5\), then \(g(0) = 5\), and in this case, the statement is false. Thus, the statement is not always true, as shown by this counterexample. The correct assessment is that the value of the function at the specific point is independent of the limit, so we cannot conclude that \(f(0)=3\) just because \(\lim_{x\rightarrow 0} f(x)=3\).

Key Concepts

Limit Definition

Understanding the concept of a limit is a linchpin in the vast field of calculus. It refers to the behavior of a function as the input approaches a particular point. In mathematical terms, the limit of a function at a certain point is the value that the function approaches as the input gets arbitrarily close to that point.

Take, for example, the expression \( \lim_{x \to a} f(x) = L \). This represents that as x moves closer and closer to 'a', the function f(x) tends to get closer to the value 'L'. Importantly, the limit tells us about the trend or direction a function is heading as we approach a point, but crucially, it does not guarantee the function actually reaches or equals 'L' at that point. This subtlety is significant and often misunderstood.

Truth Value of a Limit Statement

To further clarify, the truth value of a limit statement is contingent upon understanding the true nature of limits in calculus. If a statement declares that the limit of a function as x approaches a certain value equals L, then this only informs us about the function's behavior near that point, not necessarily at the point.

Examining the statement \( \lim_{x \to 0} f(x) = 3 \) leads us to understand that as x gets very close to 0, f(x) approaches 3. However, this limit statement does not inherently imply that \( f(0) = 3 \). The function's value at x=0 could be 3, or it could be something entirely different. Therefore, when analyzing limit statements, it is crucial to differentiate between the behavior of the function 'near' a point and its value 'at' that point. This distinction allows us to avoid misconceptions and correctly interpret mathematical assertions.

Counterexample in Limit Evaluation

The use of a counterexample can powerfully illustrate why a limit statement doesn't guarantee the function's value at a specific point. A counterexample demonstrates a situation where the initial claim does not hold, thereby disproving a general statement.

For instance, considering the function \( g(x) \) defined earlier, we showed that the function's limit as x approaches 0 is 1 regardless of g(x)'s value when x is exactly 0. By choosing different values for \( c \) when \( x=0 \) (such as 3, 5, or any other number), the limit statement remains true while the function's value at zero can vary. Hence, the existence of a single counterexample, such as \( g(0) eq 1 \) when \( c eq 1 \), is sufficient to invalidate the claim that the function's value at a point must equal the limit at that point. This logical approach solidifies our understanding and helps to correctly interpret limit statements.