Problem 7 - 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. $$ \begin{array}{ll} *\lim _{x \rightarrow 9} f(x)=6, & \lim _{x \rightarrow 6} f(x)=9, \quad f(9)=6 \\ *\lim _{x \rightarrow 9} g(x)=3, & \lim _{x \rightarrow 6} g(x)=3, \quad g(6)=9 \end{array} $$ $$ \lim _{x \rightarrow 9}(f(x)+g(x)) $$

Step-by-Step Solution

Answer

The limit is 9.

Step 1: Understand the Problem

We want to evaluate the limit \( \lim_{x \rightarrow 9} (f(x) + g(x)) \). We are given the individual limits \( \lim_{x \rightarrow 9} f(x) = 6 \) and \( \lim_{x \rightarrow 9} g(x) = 3 \). Our goal is to use these individual limits to find the limit of their sum.

Step 2: Apply Limit Sum Rule

By the property of limits, if \( \lim_{x \rightarrow c} f(x) = L \) and \( \lim_{x \rightarrow c} g(x) = M \), then \( \lim_{x \rightarrow c} (f(x) + g(x)) = L + M \). Here, \( L = 6 \) and \( M = 3 \) for \( x \to 9 \).

Step 3: Calculate the Limit

Using the limit sum rule, we calculate \( \lim_{x \rightarrow 9} (f(x) + g(x)) = 6 + 3 = 9 \).

Step 4: Verify the Information

Check if the limits deployed were appropriate and all conditions are met. Since the limits \( \lim_{x \rightarrow 9} f(x) \) and \( \lim_{x \rightarrow 9} g(x) \) exist and are finite, our use of the sum rule is valid.

Key Concepts

Limit Sum Rule

The limit sum rule is a fundamental concept in calculus that greatly simplifies the process of finding the limit of the sum of two functions. It states that if you have two functions, say \( f(x) \) and \( g(x) \), and both their limits exist as \( x \) approaches a point \( c \), then the limit of their sum is simply the sum of their limits. This can be expressed as:

• \( \lim_{x \rightarrow c} (f(x) + g(x)) = \lim_{x \rightarrow c} f(x) + \lim_{x \rightarrow c} g(x) \)

When dealing with limits, it is crucial to ensure that both individual limits exist and are finite. Only then can we apply this rule. For example, in our exercise, we know that \( \lim_{x \rightarrow 9} f(x) = 6 \) and \( \lim_{x \rightarrow 9} g(x) = 3 \). By the limit sum rule, the limit of \( f(x) + g(x) \) as \( x \) approaches 9 is \( 6 + 3 = 9 \). This rule can save time and eliminate the need for complex calculations by allowing us to directly add the known limits.

Function Evaluation

Function evaluation involves determining the output of a function for a specific input, which is particularly important when working with limits. In solving limit problems, knowing the values of \( f(x) \) and \( g(x) \) at specific points helps verify if our calculations make sense.

• Note that in the exercise, we were given specific evaluations like \( f(9) = 6 \) and \( g(9) = 3 \).

This information can be useful in confirming our understanding of the behavior of the functions around the point of interest. Even though the exact value of the function at a particular point does not affect the limit itself, it gives insight into how the function behaves. Evaluating a function at a point also provides a way to verify if errors are made in limit computations. For instance, the calculated limit matches the expected behavior suggested by evaluations, reinforcing its correctness.

Limit Properties

Limit properties are a set of rules in calculus that deal with how limits can be manipulated. They are crucial when working through more complicated limit problems, as they offer techniques to simplify different limit expressions.

• This includes properties like the limit sum rule, limit subtraction, multiplication, division, and limits of constant functions.
• For example, constants can be factored out of a limit: \( \lim_{x \rightarrow c} kf(x) = k \lim_{x \rightarrow c} f(x) \), where \( k \) is a constant.

The properties make solving and understanding limits much more approachable by providing these handy shortcuts. In our specific problem, we've applied the limit sum rule, which is one of these properties and helps simplify the sum of two functions' limits to the sum of their limits. Limit properties allow us to break down complex problems step-by-step, ensuring that we handle each part correctly before combining them into a solution. This structured approach helps in avoiding mistakes in calculations.