Problem 8 - College Algebra and Calculus: An Applied Approach

Question

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 0^{-}} \frac{\frac{1}{2+x}-\frac{1}{2}}{2 x} $$ $$ \begin{array}{|l|l|l|l|l|l|} \hline x & 0.5 & 0.1 & 0.01 & 0.001 & 0 \\ \hline f(x) & & & & & ? \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The solution involves calculating the function at particular x values, observing the resulting trend as x gets closer to '0-', estimating the limit, and then confirming the result by graphing the function. The specific values and resulting limit will depend on calculations.

Step 1: Calculate f(x) for Given x-values

To begin with, the function `f(x)` should be calculated for all given x values i.e., 0.5, 0.1, 0.01, and 0.001. This will be done by substituting these x values into the function `f(x) = (1/(2+x) - 1/2)/(2x)`.

Step 2: Fill in the Table

After calculating `f(x)` values for all x values, the results are recorded in the table. The table should now look like this: \[ \begin{array}{|l|l|l|l|l|l|} \hline x & 0.5 & 0.1 & 0.01 & 0.001 & 0 \\ \hline f(x) & f(0.5) & f(0.1) & f(0.01) & f(0.001) & ? \\ \hline \end{array} \]

Step 3: Find Limit as x Approaches 0-

Observe the pattern of `f(x)` as `x` gets closer to `0`. Use this to estimate the limit of the function as `x` approaches `0-`.

Step 4: Confirm Result with Graphing Utility

After estimating the limit, use a graphing utility to plot the function `f(x) = (1/(2+x) - 1/2)/(2x)` to confirm the result. The limit as `x` approaches `0-` should match the value observed in the table as `x` gets extremely close to `0`.

Key Concepts

Graphing Utility

A graphing utility is a tool that allows you to visualize mathematical functions and their behavior. In this exercise, it's used to confirm the limit calculated for the function. Graphing utilities can be both software applications or physical calculators. They help by:

• Providing a visual representation of the function's behavior.
• Allowing you to draw conclusions by seeing how the function behaves as it approaches a certain point.
• Offering a way to zoom into specific sections of the graph for better detail.

When using a graphing utility to confirm the limit of a function as it approaches a particular value, check the plotted graph for consistency with your calculated limit. The closer the graph's output approaches the expected limit as the input nears zero, the more confident you can be in your mathematical calculations.

Function Evaluation

Function evaluation is the process of determining the output of a function given certain input values. In the problem, you're asked to evaluate the function \[f(x) = \frac{\frac{1}{2+x} - \frac{1}{2}}{2x} \]for several x-values such as 0.5, 0.1, 0.01, and 0.001.To evaluate this function:

• Substitute the given x-value into the function.
• Perform the arithmetic operations as instructed by the function's formula.
• Record the outcome for each x-value substituting step-by-step from left to right.

This will fill up the table in the exercise and provide a set of data points that indicate how the function behaves as x approaches zero from the negative side.

Estimating Limits

Estimating limits involves observing the trend of function values as the input approaches a particular point. In this exercise, you want to find the limit of the function as x approaches zero from the left. To estimate limits:

• Calculate the function values for inputs getting progressively closer to the desired point (in this case, zero).
• Look for a pattern or an approaching value among these function outputs.
• Infer the limit from these observations.

For instance, as the x-values decrease from 0.5 to 0.001, you might see the function values stabilize or approach a particular number. This number is what you can estimate as the limit of the function as x approaches zero.

One-Sided Limits

One-sided limits focus on the behavior of a function as the input approaches a specified point from one direction—either from the left (negative direction) or the right (positive direction). In the given exercise, the limit \[\lim _{x \rightarrow 0^{-}} \frac{\frac{1}{2+x}-\frac{1}{2}}{2 x}\] refers to a one-sided limit from the left.To calculate a one-sided limit:

• Consider only the values of x that approach the specified point from the desired direction.
• Use these values to evaluate the function.
• Look for trends among the outputs as x approaches the point from the specified side.

One-sided limits can be crucial for understanding the precise behavior of a function near discontinuities or boundary points. They provide a more nuanced view of the function's behavior compared to two-sided limits, which consider approach from both directions.