Question

Use a graphing utility or spreadsheet software program to complete the table. Then use the result to estimate the limit of \(f(x)\) as \(x\) approaches infinity. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5} & 10^{6} \\ \hline f(x) & & & & & & & \\ \hline \end{array} $$ \(f(x)=\frac{x+1}{x \sqrt{x}}\)

Short answer

After performing the steps, you will get an estimate of the limit of \(f(x)\) as \(x\) approaches infinity based on the trend in the computed \(f(x)\) values. Note that the accurate value may not be determined due to the estimation.

Step-by-step solution

Fill in the table

Use a graphing utility or spreadsheet program and compute the values of \(f(x)\) for each given \(x\). The given function is \(f(x)=\frac{x+1}{x \sqrt{x}}\). So the first row of the table will contain the values of \(x\) which are \(1, 10, 100, 1000, 10000, 100000, 1000000\) (each representing \(10^{n}\), with \(n\) ranging from \(0\) to \(6\)). The second row of the table will contain the corresponding computed values of \(f(x)\).

Estimate the Limit of \(f(x)\) as \(x\) approaches infinity

To estimate the limit of \(f(x)\) as \(x\) approaches infinity, observe the trend in the computed values of \(f(x)\) as \(x\) increases, based on the filled table. If the \(f(x)\) values are approaching to a specific number as \(x\) increases, that specific number will be the limit.

Key concepts

Graphing Utility

A graphing utility is a powerful tool for visualizing mathematical concepts, particularly when dealing with functions and their behaviors. It can plot an algebraic function, allowing us to see its shape and how it behaves as the independent variable, commonly denoted as x, increases or decreases. When estimating limits, especially as x approaches infinity, graphing utilities give a visual representation that can help predict where the function’s values are headed, without exclusively relying on numerical data. By plotting the function and zooming out to see the behavior as x becomes large, students can often conclude if the function approaches a specific value, continues to increase or decrease without bound, or oscillates without settling down to a single value.

For example, with the function f(x) = (x+1)/(x√x), a graphing utility would show how the function’s values level off as x grows, hinting towards the limit. Such a visual aid complements analytical methods, giving a well-rounded approach to understanding limits at infinity.

Spreadsheet Software

Spreadsheet software, such as Microsoft Excel or Google Sheets, is invaluable for calculating and organizing numerical data. It is particularly useful in mathematics for creating tables that approximate the behavior of functions across different values. For estimating limits, one could input the algebraic expression into a cell and drag the formula down to replicate the calculation over a range of x values. As x increases, particularly towards infinity, we examine the trend in the function’s output values.

To elucidate, in the provided exercise, each power of 10 representing x is input in one column, and the corresponding function values of f(x) = (x+1)/(x√x) appear in an adjacent column. As you extend the table to larger x, you might notice the values of f(x) getting closer to a particular number, suggesting the limit as x approaches infinity. This numerical approach provides a straightforward method to gauge the function's limit and is a perfect task for spreadsheet software.

Limits at Infinity

The concept of limits at infinity in calculus refers to the value that a function approaches as the input x grows without bound. Understanding limits at infinity is crucial when analyzing the behavior of functions as they stretch towards the infinite horizons of the x-axis. It informs us about the long-term behavior or the end behavior of functions.

In the provided exercise with f(x) = (x+1)/(x√x), the limit at infinity is being sought. To estimate this limit, we observe the output of the function as x takes on extremely large values. If these outputs approach a single finite number, we can say that the limit at infinity is that number. However, it’s essential to note that not all functions will have a finite limit at infinity; some functions increase or decrease without bound, and others oscillate without approaching a finite limit. The key is to look for a pattern in the function's values and identify if they’re homing in on a particular number as x increases.

Algebraic Functions

Algebraic functions are mathematical expressions constructed using polynomial expressions, rational expressions, roots, and other algebraic operations. They are a fundamental component in the study of limits because they often make up the functions being investigated. In the context of the exercise, f(x) = (x+1)/(x√x) is a type of algebraic function known as a rational function because it is the ratio of two polynomials.

When estimating the limits of algebraic functions, particularly at infinity, it is often helpful to manipulate the expression to simplify the end behavior analysis. For instance, in the case of f(x), dividing numerator and denominator by x, the highest power of x in the denominator, simplifies the function to one that more clearly shows its behavior as x increases. Analyzing algebraic functions in this way allows a clearer insight into their limits and provides an algebraic foundation for the numerical or graphical estimations discussed previously.