Question

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

Short answer

The limit \( \lim_{x \to 6} f(g(x)) \) cannot be determined with the given information because \( \lim_{x \to 3} f(x) \) is not provided.

Step-by-step solution

Identify Inner Function and Its Limit

Here we have the composite function \( f(g(x)) \) and we are interested in the limit as \( x \to 6 \). First, identify \( g(x) \) as the inner function. We need to find \( \lim_{x \to 6} g(x) \). From the given information, this limit is \( 3 \).

Substitute the Limit of the Inner Function into the Outer Function

Now that we have \( \lim_{x \to 6} g(x) = 3 \), we need to substitute this into the outer function to find \( \lim_{y \to 3} f(y) \). Thus, we need to evaluate \( \lim_{x \to 3} f(x) \).

Evaluate Limit of the Outer Function

From the given data, there is no direct value for \( \lim_{x \to 3} f(x) \). Without additional information about \( f(x) \) at \( x = 3 \), it is not possible to determine this limit. Therefore, \( \lim_{x \to 6} f(g(x)) \) cannot be determined with the provided information.

Key concepts

Composite Functions

Understanding composite functions is a key skill in calculus and higher-level math. A composite function is created when one function is applied to the results of another function. You can think of it like a function inside another function; the output of one becomes the input of the next.
An interesting property of composite functions is how they can sometimes simplify complex relationships between variables. For instance, if you had two functions such as \( f(x) \) and \( g(x) \), a composite function could be \( f(g(x)) \). Here, \( g(x) \) acts first, and its result is then transformed by \( f(x) \).
With composite functions, limits can behave in unique ways. Often, the analysis involves carefully evaluating the limits of both the inside and outside functions. By knowing how inner and outer functions interact, mathematicians can evaluate these limits more efficiently.

Inner and Outer Functions

In composite functions, understanding the roles of the inner and outer functions is crucial. Let's take the function \( f(g(x)) \). Here, \( g(x) \) is the inner function, and \( f(x) \) is the outer function. The process involves computing the inner function first and then using its output as the input for the outer function.
Understanding which part of the composite is the inner or outer function helps in evaluating limits. It shapes how you approach the substitution process while dealing with limits.

• Start by evaluating the limit of the inner function. This is often vital because it determines the input for your outer function analysis.
• Once you establish the limit of the inner function, substitute this result into the outer function. Evaluating this new expression may lead you to the composite function’s limit.

This problem-solving strategy is foundational in calculus, enabling you to systematically tackle more complicated limit evaluations.

Limit Evaluation

Evaluating the limits of functions is fundamental in calculus. It helps us understand the behavior of functions near a particular value. When dealing with composite functions like \( f(g(x)) \), the process of limit evaluation involves multiple steps.
In our exercise, you start by assessing \( g(x) \), the inner function, as \( x \) approaches a specific value. Here, as \( x \to 6 \), \( g(x) \) approaches 3. This means you consider your focus on what happens to the outer function, \( f(x) \), as its input approaches 3.
However, you're sometimes confronted with cases where the limit cannot be determined due to missing information, like not knowing \( \lim_{x \to 3} f(x) \). Without this vital piece, the overall limit of \( \lim_{x \to 6} f(g(x)) \) remains unsolvable.

• Verify the behavior of both the inner and outer functions.
• Substitute the limit of the inner into the outer function for evaluation.
• Identify missing details about essential limits to see if further information or calculation is needed.

This meticulous approach is what makes limit evaluation such a precise and informative process, crucial for grasping deeper mathematical concepts.