Problem 34 - Precalculus

Question

Use a table of values to evaluate each function as \(x\) approaches the value indicated. If the function seems to approach a limiting value, write the relationship in words and using the limit notation. $$w(x)=\frac{x(x-2)-8}{4(x-4)} ; x \rightarrow 4$$

Step-by-Step Solution

Answer

The limit of \(w(x)\) as \(x\) approaches 4 is \(-10\): \(\lim_{{x \to 4}} w(x) = -10\).

Step 1: Set Up a Table of Values

To evaluate \(w(x)\) as \(x\) approaches 4, we create a table of values with \(x\) values close to 4, both greater and less than 4. We choose values such as 3.9, 3.99, 3.999, 4.001, 4.01, and 4.1 and calculate \(w(x)\) for each of these values.

Step 2: Substitute Values into the Function

Substitute each \(x\) value from the table into the function \(w(x) = \frac{x(x-2)-8}{4(x-4)}\) and simplify:- For \(x=3.9\): \(w(3.9) = \frac{3.9(3.9-2)-8}{4(3.9-4)}\)- For \(x=3.99\): \(w(3.99) = \frac{3.99(3.99-2)-8}{4(3.99-4)}\)- For \(x=3.999\): \(w(3.999) = \frac{3.999(3.999-2)-8}{4(3.999-4)}\)- For \(x=4.001\): \(w(4.001) = \frac{4.001(4.001-2)-8}{4(4.001-4)}\)- For \(x=4.01\): \(w(4.01) = \frac{4.01(4.01-2)-8}{4(4.01-4)}\)- For \(x=4.1\): \(w(4.1) = \frac{4.1(4.1-2)-8}{4(4.1-4)}\)

Step 3: Calculate and Analyze the Values

Calculate the values of \(w(x)\) for each \(x\):- \(w(3.9) \, \approx \, -10.25\)- \(w(3.99) \, \approx \, -10.025\)- \(w(3.999) \, \approx \, -10.0025\)- \(w(4.001) \, \approx \, -9.9975\)- \(w(4.01) \, \approx \, -9.975\)- \(w(4.1) \, \approx \, -9.75\)Observing these values, it seems that as \(x\) approaches 4, \(w(x)\) approaches \(-10\).

Step 4: State the Limit in Words and Using Limit Notation

The function \(w(x)\) approaches the value \(-10\) as \(x\) approaches 4. In limit notation, this is expressed as \(\lim_{{x \to 4}} w(x) = -10\).

Key Concepts

Function Approximation

When dealing with functions, it can be helpful to approximate the behavior around a certain point. Function approximation involves estimating the behavior of a function using nearby values to understand its pattern or tendency.
This is especially useful when the function itself becomes difficult to handle directly at a particular point. With approximation, we observe the outcomes of the function at values close to the target point to infer possible results.

• Provides insights when direct calculation at a specific point is challenging.
• Helps identify behavioral trends of the function, such as a limit it might approach.

In the context of the given exercise, function approximation aids in predicting the limit of the function as \( x \) approaches 4. This is done by computing the function at values around 4, thereby creating a clearer picture of how \( w(x) \) behaves near that point.

Table of Values

Creating a table of values is a powerful tool for visualizing a function's behavior around a certain point. By selecting values that are incrementally closer to the specified point, we can analyze how the function changes with respect to \(x\).
This method provides a straightforward way of detecting trends and approximating limits, especially when direct evaluation leads to indeterminate forms.

• Choose values close to the target to see the trend.
• Use both values greater than and less than the point to ensure a balanced view.
• Compute the function's output for each chosen value to observe patterns.

In our exercise, the table of values with \( x \) values of 3.9, 3.99, 3.999, 4.001, 4.01, and 4.1 demonstrates how \( w(x) \) approaches -10 as \( x \) gets closer to 4. This progressively detailed approach ensures that we capture any nuances in the function's behavior.

Limit Notation

Limit notation is a concise way to summarize the behavior of a function as the input approaches a specific value. This mathematical language expresses the idea that while a function might not be easy to compute directly, it approaches a certain value as its input nears the point of interest.
This concept helps capture the essence of a function's behavior in a simple statement, giving a clear result without cumbersome calculations.

• The limit is written as \( \lim_{{x \to a}} f(x) = L \), where \( a \) is the point being approached and \( L \) is the function’s limiting value.
• Useful in cases where direct substitution is not possible due to indeterminate forms.
• Allows the expression of a function's behavior at \'boundaries\' or enclosed points.

In this scenario, using limit notation, we write \( \lim_{{x \to 4}} w(x) = -10 \). This statement provides clarity and precision regarding what happens to \( w(x) \) as \( x \) moves nearer to 4.

Continuous Functions

Continuous functions are a vital concept in calculus, describing functions where small changes in the input cause only small changes in the output. This characteristic ensures there are no abrupt changes or gaps in the function's graph.
Understanding continuity serves as a foundation for evaluating limits, as it implies smoother behavior around points, and thus, helps predict how functions behave as they approach certain values.

• If a function is continuous at a point, the limit of the function as it approaches that point is simply the function's value at that point.
• Continuity avoids unexpected drops or jumps in the function's graph.
• A continuous function over an interval ensures that the function doesn’t "break" at any point within that range.

In our function \( w(x) \), analyzing its continuity around \( x = 4 \) via the table approach reveals the tendency of the function, despite being undefined exactly at 4. Since there's no jump and the trend continues smoothly, we conclude that it approaches \(-10\), demonstrating piecewise continuity in its behavior.