Problem 18 - Precalculus with Limits

Question

In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically. $$\lim_{x \to 2} \dfrac{\dfrac{1}{x+2}-\dfrac{1}{4}}{x-2}$$

Step-by-Step Solution

Answer

The limit of the function as x approaches 2 can be numerically calculated using the table of values and confirmed graphically.

Step 1: Create Table of Values

Create a table of values for the function as x gets closer and closer to 2 from both sides. This can be done by substituting values in the function close to 2, like 1.9, 1.99, 1.999 from the left and 2.1, 2.01, 2.001 from the right.

Step 2: Estimate the Limit

Approximate the limit of the function as x approaches 2, using the calculated table of values. Observe the y-values; as x gets closer and closer to 2, the y-values should get closer to a certain value. That value will be the limit.

Step 3: Graph the Function

Graph the function using a graphing utility. Set the x-axis limits so that they surround the value of 2 and observe where the function value (y) is heading as x gets closer to 2. The value should validate the estimated limit from step 2.

Key Concepts

Numerical Estimation

Numerical estimation is a powerful method to approximate the limit of a function when approaching a particular value. In this exercise, we're focused on estimating \[ \lim_{x \to 2} \frac{\frac{1}{x+2}-\frac{1}{4}}{x-2} \]bub using x-values close to 2. To perform numerical estimation, begin by selecting numbers very near 2 from both left (e.g., 1.9, 1.99, 1.999) and right (e.g., 2.1, 2.01, 2.001). This process is known as approaching from both sides.

By substituting these values into the given function, you can observe how the outputs (or y-values) behave. If the function is continuous around the point of interest, these y-values will get closer to a specific number. This number is the estimated limit. It helps to refine your numerical estimation by selecting closer x-values if necessary.

Steps to perform numerical estimation effectively:

• Choose x-values approaching the target value from both sides.
• Substitute these x-values into the function.
• Observe the resulting y-values for a pattern or trend.
• Adjust and select closer x-values if needed for precision.
• Identify the number y-values converge to as the estimated limit.

Table of Values

A table of values is a systematic way to record how a function behaves as x approaches a specific point. For this exercise, a table of values will help us assess \[ \lim_{x \to 2} \frac{\frac{1}{x+2}-\frac{1}{4}}{x-2} \].The point is to get a clearer picture of what happens in the behavior of the function.

To set up a table of values:

• Select x-values approaching the point of interest. In this case, choose numbers smaller and larger than 2, like 1.9, 1.99, 1.999 and 2.1, 2.01, 2.001.
• Calculate the corresponding y-values by substituting these x-values into the function.
• Record these x and y pairs in the table to easily visualize trends.

This table will provide an intuitive understanding of how the y-values behave as x gets closer to 2 from both sides. If the y-values tend to settle around a particular number, you have a strong basis for your limit estimation.

Graphing Utility

A graphing utility, such as a graphing calculator or software, serves as a visual confirmation of findings from numerical estimation and table of values. For the function \[ \lim_{x \to 2} \frac{\frac{1}{x+2}-\frac{1}{4}}{x-2} \], this tool is invaluable.

By graphing the function, you can see the function's curve and how it behaves near x = 2. Here's how a graphing utility enhances your understanding:

• Set up the graph with x-axis limits narrowly surrounding x = 2.
• Observe the trend of the curve as it approaches x = 2. Note where the y-value appears to converge.
• Compare the graphically observed limit with the numerically estimated limit.

By utilizing a graphing utility, you not only confirm your numerical and tabular estimates but also gain a deeper visual intuition of the limit concept. It ensures that all methods point towards the same conclusion, allowing for a reliable and comprehensive understanding.