Problem 12 - Calculus of a Single Variable

Question

In Exercises 7–14, create a table of values for the function and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 2} \frac{[x /(x+1)]-(2 / 3)}{x-2} $$

Step-by-Step Solution

Answer

The limit of the function as x approaches 2 is determined by observation from both a table of values and a graphical representation. In order to provide a precise answer, actual calculations should be made using, for example, a calculator or suitable mathematical software.

Step 1: Create a Table of Values

Pick values that are close to 2 from both the left and the right. Then substitute these values into the function. For instance, take 1.9, 1.99, 1.999, 2.001, 2.01, and 2.1.

Step 2: Calculate the Corresponding Function Values

Substitute the chosen values from Step 1 into the function to calculate the approximated function output. It's suggested to use a calculator for this step due to the complexity of the function.

Step 3: Estimate the Limit

Analyze the function outputs as x gets closer and closer to 2 from both the left and the right. If these outputs tend to approach a certain value, this is the estimated limit.

Step 4: Graph the Function

Use a graphing utility to graph the function. Verify whether the graph approaches the same value as x get closer to 2 from both left and right, consistent with the estimated limit from the table of values. If these results agree, the limit estimation can be considered accurate.

Key Concepts

Understanding Calculus in Limit Problems

Calculus is a branch of mathematics that focuses on the study of change. One of its fundamental concepts is the limit. A limit describes the behavior of a function as the input approaches a particular point. In the exercise above, we're looking at how the function behaves as \( x \) approaches 2. This is critical in understanding the overall continuity and differentiability of functions. It's important to note that limits are foundational in calculus because they allow us to comprehend and describe instantaneous rates of change and areas under curves. Understanding these concepts can significantly improve your ability to tackle problems involving derivative and integral calculus.

Utilizing Graphing Utilities for Visualization

A graphing utility is a powerful tool that helps visualize complex functions and their behavior. For our function \( \lim _{x \rightarrow 2} \frac{[x /(x+1)]-(2 / 3)}{x-2}\), a graphing utility can confirm our limit estimation by providing a visual representation.

• These utilities can graph both simple and complicated functions.
• Visualization can show whether a function approaches a particular value as \( x \) approaches our limit of interest, 2, in this case.
• They provide a cross-check against analytical methods, like table of values, ensuring our reasoning and calculations are correct.

Graphing utilities often come with features to zoom and closely examine specific parts of the graph, making it even easier to observe the behavior near the point of interest.

Creating and Analyzing a Table of Values

A table of values is a simple yet effective method to approximate the limit. By substituting values close to the target approach point, we can observe trends in the function's output. Here’s how to go about it:

• Choose values for \( x \) that are close to 2, like 1.9, 1.99, and so on.
• Substitute these values into the function and calculate the corresponding outputs.
• Observe the values of the function as they get closer to the limit point. Does it seem like they're converging to a single number?

This process helps give an empirical estimation of the limit, building a strong foundation for your understanding of the limiting process. It's a hands-on way to verify theoretical findings and gives a tangible feel for what limits do in practical instances.

Estimating Limits: Observing Trends and Patterns

Limit estimation involves observing how a function behaves as the input approaches a certain value. Here, we examine the function as \( x \) gets closer to 2 from both sides. To perform a proper limit estimation, note the following:

• Analyze whether the function outputs from your table of values converge to a specific number.
• Use this estimation to predict the behavior of a function near that point.
• Correlate these findings with the graph derived from a graphing utility.

If you notice that the function values stabilize towards a particular number as \( x \) tends towards 2, you have a strong candidate for the limit. This method validates both theoretical insights from calculus and practical evidence from numerical and graphical observations.