Problem 15 - 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. $$\lim_{x \to 0} \dfrac{\sqrt{x+5} - \sqrt{5}}{x}$$

Step-by-Step Solution

Answer

The short answer will be the value that the function seems to be approaching as \(x\) gets closer to zero, from both the left and right side. The exact value will depend on the values in the table and the graph.

Step 1 - Create a table of values

A table of values could be created for \(x\) values approaching zero both from the left and right. As per the function \(\frac{{\sqrt{x+5} - \sqrt{5}}}{x}\), plug in the \(x\) values into the function to get corresponding function values.

Step 2 - Analyze the table

Look at the trends in the table values. As \(x\) gets closer to 0, are the \(y\) values getting closer to a certain value? That would indicate the limit of the function as \(x\) approaches 0.

Step 3 - Use a graphing utility

Input the function \(\frac{{\sqrt{x +5} - \sqrt{5}}}{x}\) into a graphing utility to visualize the behavior of the function as \(x\) approaches 0. Confirm if the graphic representation of the function supports the findings from the table.

Step 4 - Drawing Conclusions

Draw a conclusion on the limit of the function as \(x\) approaches 0 based on the table of values and the graph. If both of these agree about the limit, then your answer is likely correct.

Key Concepts

Function Values Table

A function values table is a powerful tool for estimating limits numerically. To start, you must select a range of input values for \(x\) that approach the point of interest—in this case, zero—from both the left and right. These values might include, for example, -0.1, -0.01, -0.001, 0.001, 0.01, and 0.1. For each of these \(x\) values, substitute them into the function \(\frac{\sqrt{x+5} - \sqrt{5}}{x}\) to calculate the corresponding \(y\) values.

This process helps us to see a pattern or trend in the \(y\) values. As the \(x\) values get increasingly closer to zero, observe if the \(y\) values converge towards a specific number. This number is a hint towards estimating the function's limit. It's a method that combines mathematical calculation with observational skills.

Graphing Utility

A graphing utility, such as a graphing calculator or software, allows us to visualize functions in a dynamic way. By inputting the equation \(\frac{\sqrt{x +5} - \sqrt{5}}{x}\) into the graphing utility, you can see a graphical representation of how the function behaves as \(x\) approaches zero.

This visual approach complements the numerical analysis from the function values table by offering a clearer picture. You should look for a trend in the graph that matches what you observed in your table of values. If the graph appears to level off to a certain value as \(x\) nears zero, this further supports your estimation of the limit. This alignment between numerical and graphical data is crucial for confirming your hypothesis.

Limit Estimation

Limit estimation is the process of finding the value that a function approaches as the input nears a specific point. In this exercise, we aim to estimate \(\lim_{x \to 0} \frac{\sqrt{x+5} - \sqrt{5}}{x}\). Combining both qualitative and quantitative analysis helps ensure the accuracy of your limit prediction.

From the function values table, you infer the limit by observing where the \(y\) values converge as \(x\) gets closer to zero. With the graphing utility, you check if the visual behavior of the graph around \(x = 0\) supports the same conclusion. Together, these approaches provide a well-rounded strategy for estimating limits. Interactive tools and calculated tables thereby work hand-in-hand to bolster your understanding of how functions behave at particular points.