Problem 91 - 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, explain why or give an example to show why it is false. If \(\lim _{x \rightarrow 2} f(x)=3\) and \(\lim _{x \rightarrow 2} g(x)=0\), then \(\lim _{x \rightarrow 2}[f(x)] /[g(x)]\) does not exist.

Step-by-Step Solution

Answer

The statement is true. The limit of the quotient \(\lim_{x \rightarrow 2} \frac{f(x)}{g(x)}\) does not exist because we are dividing by zero, which is undefined. Using the limit laws for quotients, we find that \(\lim_{x \rightarrow 2} \frac{f(x)}{g(x)} = \frac{\lim_{x \rightarrow 2} f(x)}{\lim_{x \rightarrow 2} g(x)} = \frac{3}{0}\), and since division by zero is undefined, the limit does not exist.

Step 1: Analyze the given limits

We are given: 1. \(\lim_{x \rightarrow 2} f(x) = 3\) 2. \(\lim_{x \rightarrow 2} g(x) = 0\)

Step 2: Determine the limit of the quotient

We wish to find the limit of the quotient as x approaches 2: \(\lim_{x \rightarrow 2} \frac{f(x)}{g(x)}\)

Step 3: Apply limit laws for quotients

Using the limit laws for quotients, we have \(\lim_{x \rightarrow 2} \frac{f(x)}{g(x)} = \frac{\lim_{x \rightarrow 2} f(x)}{\lim_{x \rightarrow 2} g(x)}\) Now, we can substitute the given limits of f(x) and g(x) into the equation: \(\lim_{x \rightarrow 2} \frac{f(x)}{g(x)} = \frac{3}{0}\)

Step 4: Discuss the result

Since dividing by zero is undefined, the limit of the quotient does not exist. Therefore, the statement "If \(\lim _{x \rightarrow 2} f(x)=3\) and \(\lim _{x \rightarrow 2} g(x)=0\), then \(\lim _{x \rightarrow 2}[f(x)] /[g(x)]\) does not exist." is true.

Key Concepts

Limit Laws

Limit laws are fundamental tools in calculus used to calculate the limit of a function based on limits of simpler components. They allow us to handle more complex expressions by breaking them down into simple, manageable pieces. For example, the limit of a sum is the sum of the limits, as long as both limits exist. This same idea extends to products, differences, and quotitive operations.

• Sum Law: \(\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)\)
• Product Law: \(\lim_{x \to a} [f(x) \, g(x)] = \lim_{x \to a} f(x) \, \lim_{x \to a} g(x)\)
• Quotient Law: \(\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}; \text{ if } \lim_{x \to a} g(x) eq 0\)

Limit laws are indispensable when evaluating limits analytically, and time spent mastering these laws is time saved solving more complex calculus problems.

Indeterminate Forms

Indeterminate forms in calculus often arise when evaluating limits that don't produce a clear or direct result. The form \(\frac{0}{0}\) occurs when both the numerator and denominator of a function approach zero, making the direct calculation of the limit impossible without further manipulation or simplification.
In our specific exercise, when evaluating the limit \(\lim_{x \to 2} \frac{f(x)}{g(x)}\), the denominator approaches zero, which leads to the indeterminate form \(\frac{3}{0}\). This is undefined in mathematics because division by zero is not possible. However, it is important to distinguish between limits resulting in forms like \(\frac{0}{0}\) or fixed constants over zero, which usually indicate the presence of a vertical asymptote or other peculiarity in the behavior of the function.

• If encountered, such forms often require advanced techniques like L'Hôpital's Rule or algebraic manipulation to resolve.

Recognizing indeterminate forms is crucial for determining when and how to apply additional calculus techniques to find limits.

Calculus Problem Solving

Solving calculus problems involves a strategic method to break down and analyze expressions and limits using various rules and laws. Let's discuss the method typically employed in solving limit problems like the one given in the exercise.
First, identify given expressions and their approaching limits. Then, analyze which limit laws apply. When you come across a situation that leads to an indeterminate form, like \(\frac{3}{0}\), acknowledge the lack of an explicit limit. This leads you to either state that the limit does not exist or take further steps if context allows more advanced techniques.

• Break down complex expressions into recognizable forms.
• Apply the correct limit laws sequentially and logically.
• Recognize special cases such as indeterminate forms.
• Seek further manipulations if initial evaluations return undefined forms.

Approaching calculus with clear step-by-step problem solving not only helps in finding solutions but also builds a deep understanding of mathematical concepts, enabling you to tackle more intricate calculus challenges with confidence.