Problem 14 - APEX Calculus

Question

Use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not. $$ \begin{array}{ll} *\lim _{x \rightarrow 9} f(x)=6, & \lim _{x \rightarrow 6} f(x)=9, \quad f(9)=6 \\ *\lim _{x \rightarrow 9} g(x)=3, & \lim _{x \rightarrow 6} g(x)=3, \quad g(6)=9 \end{array} $$ $$ \lim _{x \rightarrow 6} f(x) g(x)-f^{2}(x)+g^{2}(x) $$

Step-by-Step Solution

Answer

The limit is -45.

Step 1: Understanding the Problem

We need to evaluate the limit as \( x \) approaches 6 for the expression \( f(x)g(x) - f^2(x) + g^2(x) \). We will apply the limits of individual parts of the expression based on the information provided.

Step 2: Apply Limits Individually

We apply limits to each component of the expression: \( \ \lim_{x \to 6} f(x)g(x) = \lim_{x \to 6} f(x) \cdot \lim_{x \to 6} g(x) = 9 \cdot 3 = 27 \ \lim_{x \to 6} f^2(x) = (\lim_{x \to 6} f(x))^2 = 9^2 = 81 \ \lim_{x \to 6} g^2(x) = (\lim_{x \to 6} g(x))^2 = 3^2 = 9 \).

Step 3: Substitute and Simplify the Limits

Substitute the obtained limits into the expression and simplify: \( \lim_{x \to 6} (f(x)g(x) - f^2(x) + g^2(x)) = 27 - 81 + 9 \).

Step 4: Compute the Result

Calculate the final result: \( 27 - 81 + 9 = -45 \). This is the value of the limit expression as \( x \) approaches 6.

Key Concepts

Evaluating Limits

Evaluating limits is a fundamental part of calculus. It involves finding the value that a function approaches as the input (usually denoted as \( x \)) nears a particular point. In our exercise, evaluating the limit means determining the value of the function expression \[\lim _{x \rightarrow 6} (f(x)g(x) - f^{2}(x) + g^{2}(x)).\]Here's how to approach problems like this:

• Identify which point \( x \) is approaching. In this example, \( x \) approaches 6.
• Break the expression into parts and apply known limits to each part.
• Substitute the limits and perform arithmetic operations to simplify the expression.

By following these steps, you can effectively simplify and solve limit problems.

Limit Laws

Limit laws are essential tools that simplify the process of evaluating limits by allowing the combination of individual limits. They are particularly helpful in dealing with expressions composed of several functions. In our example, these laws help us evaluate the components of the expression separately:1. **Product Law:** If \( \lim_{x \to a} f(x) = L \) and \( \lim_{x \to a} g(x) = M \), then \( \lim_{x \to a} (f(x) \cdot g(x)) = L \cdot M \).
This allows us to find the limit of the product of functions.2. **Power Law:** If \( \lim_{x \to a} f(x) = L \), then \( \lim_{x \to a} (f(x))^n = L^n \).
This applies to expressions where a function is raised to a power.3. **Sum and Difference Law:** If \( \lim_{x \to a} f(x) = L \) and \( \lim_{x \to a} g(x) = M \), then \( \lim_{x \to a} (f(x) \pm g(x)) = L \pm M \).By applying these laws to the parts of our expression, each individual limit is simplified, making the evaluation of the entire expression straightforward.

Functions in Calculus

Functions are the backbone of calculus and come in many forms, such as linear, quadratic, polynomial, and trigonometric. Understanding functions is crucial for evaluating limits, as they describe the behavior of expressions as \( x \) approaches a certain value.In our exercise, we deal with two functions \( f(x) \) and \( g(x) \). Each function has known limits as \( x \) approaches specific points:

• \( \lim _{x \to 9} f(x)=6 \) and \( \lim _{x \to 6} f(x)=9 \).
• \( \lim _{x \to 9} g(x)=3 \) and \( \lim _{x \to 6} g(x)=3 \).
• The values of the functions at the points \( f(9)=6 \) and \( g(6)=9 \) indicate their output exactly at those points.

Knowing these values and limits allows us to substitute them into expressions confidently, applying limit laws and concepts to find solutions. Mastery of functions and their limits is fundamental to solving more complex calculus problems.