Question

Numerical and Graphical Analysis Use a graphing utility to complete the table for each function and graph each function to estimate the limit. What is the value of the limit when the power of \(x\) in the denominator is greater than 3\(?\) $$ \begin{array}{|c|c|c|c|c|c|c|c|}\hline x & {1} & {0.5} & {0.2} & {0.1} & {0.01} & {0.001} & {0.0001} \\ \hline f(x) & {} & {} & {} & {} & {} \\\ \hline\end{array} $$ $$ \begin{array}{ll}{\text { (a) } \lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x}} & {\text { (b) } \lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x^{2}}} \\\ {\text { (c) } \lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x^{3}}} & {\text { (d) } \lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x^{4}}}\end{array} $$

Short answer

The value of the limit when the power of \(x\) in the denominator is greater than 3 tends to 0.

Step-by-step solution

Analyze and Graph the Functions

To start with, analyze and graph the given functions. This can be performed graphically by plotting the functions using a graphing utility. The functions to be plotted are (a) \(\frac{x-\sin x}{x}\), (b) \(\frac{x-\sin x}{x^{2}}\), (c) \(\frac{x-\sin x}{x^{3}}\) and (d) \(\frac{x-\sin x}{x^{4}}\)

Complete Table using the results from the graph and calculation

Now, refer to the graph to complete the table by substituting the values of x in the respective functions (a) \(\frac{x-\sin x}{x}\), (b) \(\frac{x-\sin x}{x^{2}}\), (c) \(\frac{x-\sin x}{x^{3}}\) and (d) \(\frac{x-\sin x}{x^{4}}\). For x as 1, 0.5, 0.2, 0.1, 0.01, 0.001, and 0.0001, compute the corresponding function values by plugging into the respective functions.

Estimation of Limit

After obtaining all function values corresponding to x, estimate the limit from the table for each of the functions. Further, analyze the behaviour for the value of the limit of functions when power of x in the denominator is greater than 3. The approach will be applied to estimate the value of the limit for (c) \(\frac{x-\sin x}{x^{3}}\) and (d) \(\frac{x-\sin x}{x^{4}}\)

Key concepts

Graphing Utility

A graphing utility is a powerful tool that helps in visualizing mathematical functions by plotting their corresponding graphs. These utilities are great for providing a visual representation of how functions behave and can aid in understanding complex relationships within the realm of calculus and algebra.

When using a graphing utility, the user inputs a function, and the tool generates the graph according to the specified parameters. In this exercise, you might use a graphing utility like Desmos or a graphing calculator to plot functions such as \( \frac{x-\sin x}{x} \), \( \frac{x-\sin x}{x^2} \), \( \frac{x-\sin x}{x^3} \), and \( \frac{x-\sin x}{x^4} \).

Graphing utilities can provide insight into limits, allowing for a visual estimation of a function's behavior as \( x \) approaches a particular value. This is crucial in graphical analysis and can guide the understanding of limits in calculus.

Numerical Analysis

Numerical analysis involves mathematically calculating values to understand a particular function's behavior. This approach typically includes creating a table of values by substituting different \( x \)-values into the function, as seen with the given exercise.

For example, you substitute values like 1, 0.5, 0.2, 0.1, 0.01, 0.001, and 0.0001 into the functions \( \frac{x-\sin x}{x} \), \( \frac{x-\sin x}{x^2} \), \( \frac{x-\sin x}{x^3} \), and \( \frac{x-\sin x}{x^4} \).

By filling in the table, you perform a numerical analysis that helps in estimating the limit as \( x \) approaches 0 from the positive side. This step-by-step calculation clarifies the behavior of each function, especially when \( x \) becomes very small.

Graphical Analysis

Graphical analysis is the method of using graphical representations derived from graphing utilities to analyze the behavior of mathematical functions. It is a visual approach to understanding how changes in variables affect the shape and position of the graph.

In the context of this exercise, graphical analysis assists in estimating limits when \( x \) is approaching 0. By observing how the graphs of \( \frac{x-\sin x}{x} \), \( \frac{x-\sin x}{x^2} \), \( \frac{x-\sin x}{x^3} \), and \( \frac{x-\sin x}{x^4} \) behave as \( x \) gets smaller, you can make inferences about their limits.

Visual analysis reveals trends and asymptotic behavior that might not be obvious from looking at numerical data alone. This can help identify patterns, such as whether the function approaches a particular limit or remains undefined.

Functions

Functions are mathematical entities that describe the relationship between sets of inputs and their corresponding outputs. Understanding the behavior of functions is essential for studying calculus and analyzing limits.

In this exercise, the focus is on rational functions formed by \( \frac{x-\sin x}{x^n} \) where \( n \) is a positive integer. These types of functions can exhibit interesting behaviors as the denominator's power increases. For example:

• When \( n = 1 \), the function behaves uniquely compared to when \( n > 1 \).
• Different powers in the denominator lead to constraints that affect the function's limits and asymptotic behavior.
• The function's limit can also reach different values or become undefined as the power grows beyond 3.

Exploring the power of \( x \) in the denominator through graphical and numerical analysis helps solidify understanding of how functions interact and approach limits.