Problem 13 - 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. $$ \begin{array}{ll} *\lim _{x \rightarrow 9} f(x)=6, & \lim _{x \rightarrow 6} f(x)=9, \quad f(9)=6 \\ *\lim _{x \rightarrow 9} g(x)=3, & \lim _{x \rightarrow 6} g(x)=3, \quad g(6)=9 \end{array} $$ $$ \lim _{x \rightarrow 6} g(f(f(x))) $$

Step-by-Step Solution

Answer

The limit is 3.

Step 1: Evaluate \\(\lim_{x \rightarrow 6} f(x)\\)

Given that \(\lim_{x \rightarrow 6} f(x) = 9\)\, we know that as \(x\) approaches 6, \(f(x)\) approaches 9.

Step 2: Evaluate \\(\lim_{x \rightarrow 6} f(f(x))\\)

Since \(f(x)\) approaches 9 as \(x\) approaches 6, we need to find \(\lim_{x \rightarrow 9} f(x)\). It is given that \(\lim_{x \rightarrow 9} f(x) = 6\). Therefore, \(\lim_{x \rightarrow 6} f(f(x)) = 6\).

Step 3: Evaluate \\(\lim_{x \rightarrow 6} g(f(f(x)))\\)

Now that we know \(\lim_{x \rightarrow 6} f(f(x)) = 6\)\, we need to find \(\lim_{x \rightarrow 6} g(y)\) when \(y ightarrow 6\). From the given information, \(\lim_{x \rightarrow 6} g(x) = 3\). Thus, \(\lim_{x \rightarrow 6} g(f(f(x))) = 3\).

Key Concepts

Composite Functions

Composite functions are combinations of two or more functions, where the output of one function becomes the input of another. For example, if you have two functions, \(f(x)\) and \(g(x)\), their composite function is written as \(g(f(x))\). In this case, you first apply \(f\), and then apply \(g\) to the result. This composition can be crucial in multistage processes where each stage relies on the results from the previous one.
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To evaluate a composite function, like in our exercise, you first resolve the innermost function towards its limit. Next, substitute this result into the next function in the composition and continue until all functions are evaluated. This ensures composite functions are analyzed systematically, capturing each function's behavior as the input changes.
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When working with composite functions, remember:

• The order of operations matters – the first function must be resolved before moving to the next.
• Each function acts as a separate step in a larger calculation pathway.

Composite functions help understand complex systems by breaking them into simpler functional steps.

Evaluating Limits

Evaluating limits involves finding the value that a function approaches as the input approaches a particular point. In our example, the task is to find the limit of a composite function \(g(f(f(x)))\) as \(x\) approaches a specific value.
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When evaluating limits, a step-by-step approach is useful, especially in the context of nested functions:

• First, evaluate the innermost function's limit (\(\lim_{x \rightarrow 6} f(x)\)).
• Use the result to find the limit of the next function \(f(f(x))\), turning it into \(\lim_{x \rightarrow 9} f(x)\), given that \(f(x)\) approaches 9.
• Finally, fit this result into the outermost function \(g(f(f(x)))\) and resolve \(\lim_{x \rightarrow 6} g(x)\) to obtain the solution.

The key to evaluating limits of composite functions is handling each function one step at a time. This prevents mixing up different function behaviors under various input conditions.

Continuity in Calculus

Continuity is a fundamental concept in calculus that ensures a function behaves predictably near a given point. A function is continuous at a specific point if there are no sudden jumps or breaks, and it can be drawn without lifting a pencil.
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In the context of limits, continuity implies that the limit of a function as \(x\) approaches a given point equals the function's value at that point. This property simplifies evaluating limits, as you can directly substitute the point into the function if continuity is confirmed.
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However, if a function exhibits discontinuity—like a jump or hole—it requires careful analysis of the limit. Sometimes, you'll need different approaches, such as left-hand and right-hand limits, to understand the behavior fully.
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From our example:

• \(f(x)\) and \(g(x)\) were analyzed for limits approaching specific values, emphasizing their continuity.
• The task relied on understanding that these functions behaved regularly at said points, without breaks.

Continuous functions allow for easier manipulation in calculus, aiding in solving more complex mathematical problems effectively.