Problem 47 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indicated limit given that \(\lim _{x \rightarrow a} f(x)=3\) and \(\lim _{x \rightarrow a} g(x)=4\) \(\lim _{x \rightarrow a} \frac{2 f(x)-g(x)}{f(x) g(x)}\)

Step-by-Step Solution

Answer

The short answer is: \(\lim _{x \rightarrow a} \frac{2 f(x)-g(x)}{f(x) g(x)} = \frac{1}{6}\).

Step 1: Recall basic limit properties

Recall the following basic limit properties: 1. \(\lim _{x \rightarrow a} [c * f(x)] = c * \lim _{x \rightarrow a} f(x)\), where c is a constant. 2. \(\lim _{x \rightarrow a} [f(x) \pm g(x)] = \lim _{x \rightarrow a} f(x) \pm \lim _{x \rightarrow a} g(x)\) 3. \(\lim _{x \rightarrow a} [f(x) * g(x)] = \lim _{x \rightarrow a} f(x) * \lim _{x \rightarrow a} g(x)\) 4. \(\lim _{x \rightarrow a} \frac{f(x)}{g(x)} = \frac{\lim _{x \rightarrow a} f(x)}{\lim _{x \rightarrow a} g(x)}\), provided \(\lim _{x \rightarrow a} g(x) \neq 0\).

Step 2: Apply limit properties to the given function

We are given that \(\lim _{x \rightarrow a} f(x)=3\), and \(\lim _{x \rightarrow a} g(x)=4\). We want to find the limit of the function \(\lim _{x \rightarrow a} \frac{2 f(x)-g(x)}{f(x) g(x)}\). Using the basic limit properties mentioned in Step 1, we can break down the limit of the given function: \(\lim _{x \rightarrow a} \frac{2 f(x)-g(x)}{f(x) g(x)} = \frac{\lim _{x \rightarrow a} (2 f(x)-g(x))}{\lim _{x \rightarrow a} (f(x) g(x))}\) We can further break down the limit: \(\lim _{x \rightarrow a} (2 f(x)-g(x)) = 2 \lim _{x \rightarrow a} f(x) - \lim _{x \rightarrow a} g(x)\) \(\lim _{x \rightarrow a} (f(x) g(x)) = \lim _{x \rightarrow a} f(x) \cdot \lim _{x \rightarrow a} g(x)\) Now, plug the given limits for f(x) and g(x) into the equation: \(\lim _{x \rightarrow a} \frac{2 f(x)-g(x)}{f(x) g(x)} = \frac{2(3) - 4}{3 \cdot 4}\)

Step 3: Simplify and find the limit

Now we simply need to simplify our expression and find the limit: \(\lim _{x \rightarrow a} \frac{2 f(x)-g(x)}{f(x) g(x)} = \frac{6 - 4}{12} = \frac{2}{12}\) Simplifying our fraction, we can get: \(\lim _{x \rightarrow a} \frac{2 f(x)-g(x)}{f(x) g(x)} = \frac{1}{6}\) So, the requested limit is \(\boxed{\frac{1}{6}}\).

Key Concepts

Limits

Understanding limits is a foundational concept in calculus. It involves evaluating what value a function approaches as the input approaches a certain point. Consider the notation \( \lim_{x \to a} f(x) \), which reads as "the limit of \( f(x) \) as \( x \) approaches \( a \)." Here, the focus is on identifying the behavior of \( f(x) \) near \( x = a \), even if the function isn't defined right at \( a \).
This is powerful for dealing with indeterminate forms or discontinuities. Limits help us make sense of functions where direct evaluation might not be straightforward, ensuring continuity and defining derivatives and integrals.

Limit Properties

Limit properties simplify the process of evaluating limits by providing easy-to-follow rules. There are several key properties:

• Constant Factor: \( \lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x) \), where \( c \) is a constant. This means constants can be pulled out.
• Sum/Difference: \( \lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x) \). You can split up sums and differences easily.
• Product: \( \lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) \). Multiply limits individually.
• Quotient: \( \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \), provided the denominator doesn’t approach zero. This is helpful for fractions.

These properties make complex expressions much more manageable by breaking them into simpler components.

Functions

A function is a relation between a set of inputs and a set of permissible outputs. In calculus, functions are typically represented as \( f(x) \) and analyzed for various behaviors, such as continuity, differentiability, and integrability.
They are the core building blocks for evaluating limits, derivatives, and integrals. Functions can be polynomials, exponential, logarithmic, and many others. Understanding how to manipulate and evaluate functions is crucial for solving calculus problems effectively.

Rational Functions

Rational functions are fractions involving polynomials, expressed as \( \frac{P(x)}{Q(x)} \) where \( P \) and \( Q \) are polynomials. These can have interesting behaviors, such as vertical asymptotes and holes, caused by the zeros of the denominator, \( Q(x) \).
When evaluating limits, identifying and addressing these behaviors is key.
In the given exercise, understanding rational functions helps to break down complex limits with their properties, ensuring we manage both the numerator and the denominator correctly to find the limit.