Problem 4 - Calculus and its applications

Question

Classify each statement as either true or false. If \(\lim _{x \rightarrow 4} F(x)=7,\) then \(\lim _{x \rightarrow 4}[c \cdot F(x)]=7 c\)

Step-by-Step Solution

Answer

True. The statement follows the constant multiple rule of limits.

Step 1: State the Given Limit

We are given that \(\lim _{x \rightarrow 4} F(x) = 7\). This tells us the behavior of the function \(F(x)\) as \(x\) approaches \(4\), specifically that \(F(x)\) approaches the value \(7\).

Step 2: Apply the Constant Multiple Rule of Limits

According to the properties of limits, if the limit of \(F(x)\) as \(x\) approaches a given point exists, then the limit of \(c \cdot F(x)\) as \(x\) approaches the same point \(x\) also exists and is \(c\) times the original limit. Mathematically, \(\lim_{x \rightarrow 4} [c \cdot F(x)] = c \cdot \lim_{x \rightarrow 4} F(x)\).

Step 3: Calculate the Limit with the Constant

Now we substitute the known limit into the formula: \(\lim_{x \rightarrow 4} [c \cdot F(x)] = c \cdot 7 = 7c\). So, the expression \(\lim _{x \rightarrow 4}[c \cdot F(x)] = 7c\) is correctly calculated.

Step 4: Conclusion on Statement Classification

The statement is true because using the constant multiple limit rule, \(\lim _{x \rightarrow 4}[c \cdot F(x)]=c \cdot 7=7c\), which matches the statement given in the exercise.

Key Concepts

Constant Multiple Rule

The constant multiple rule is a fundamental principle in calculus concerning limits. When dealing with limits, if you have a function multiplied by a constant, you can confidently say that the limit of this operation exists. If the limit of function \(F(x)\) as \(x\) approaches a point \(a\) is \(L\), the constant multiple rule states that \(\lim_{x \to a} [c \cdot F(x)] = c \cdot L\). For example, if \(\lim_{x \to 4} F(x) = 7\), then \(\lim_{x \to 4} [c \cdot F(x)] = c \cdot 7 = 7c\). By using this rule, we see how the functions scale with multiplication by a constant, ensuring the limit maintains the multiplication factor of the constant factor.

Limit Properties

Limit properties streamline the process of evaluating limits in calculus. Let's unpack some of these useful properties.

• **Sum Rule**: \(\lim_{x \to a} [F(x) + G(x)] = \lim_{x \to a} F(x) + \lim_{x \to a} G(x)\)
• **Difference Rule**: \(\lim_{x \to a} [F(x) - G(x)] = \lim_{x \to a} F(x) - \lim_{x \to a} G(x)\)
• **Product Rule**: \(\lim_{x \to a} [F(x) \, G(x)] = \lim_{x \to a} F(x) \, \lim_{x \to a} G(x)\)
• **Quotient Rule**: \(\lim_{x \to a} \left[ \frac{F(x)}{G(x)} \right] = \frac{\lim_{x \to a} F(x)}{\lim_{x \to a} G(x)} \), given \(\lim_{x \to a} G(x) eq 0\)

Together, these properties allow us to manipulate complex expressions, simplifying the evaluation and making it easier to find the limit of a combined operation. The constant multiple rule is one such property, allowing simplification when a constant factor is present outside a function.

Function Behavior

Understanding the behavior of a function as it approaches a specific point is central to graph analysis and predicting future trends in calculus. The behavior can differ greatly depending on the function's nature.

• A function may approach a specific value, illustrating convergence, as we see when \(F(x)\) approaches 7 as \(x\) nears 4.
• Alternatively, a function might exhibit oscillation, where it moves back and forth around a point, never settling on a single value, often leading to no limit existing.
• Functions can also approach infinity, indicating divergence if they increase or decrease without bound as \(x\) approaches a certain point.

In our example, the statement \(\lim_{x \to 4} F(x) = 7\) tells us that \(F(x)\) converges to 7 as \(x\) nears 4, illustrating this specific behavior. Recognizing these behaviors allows for more precise analysis and better understanding of how functions act near certain points.