Problem 50 - Essential Calculus

Question

Numerical, Graphical, and Analytic Analysis In Exercises \(49-52,\) use a graphing utility to complete the table and estimate the limit as \(x\) approaches infinity. Then use a graphing utility to graph the function and estimate the limit. Finally, find the limit analytically and compare your results with the estimates. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5} & 10^{6} \\ \hline f(x) & & & & & & & \\ \hline \end{array} $$ $$ f(x)=x^{2}-x \sqrt{x(x-1)} $$

Step-by-Step Solution

Answer

The limit of the function \(f(x)=x^{2}-x \sqrt{x(x-1)}\) as \(x\) approaches infinity, after numerical, graphical, and analytic analysis, will be 0. The function approaches the constant 0 as we increase \(x\), which is also evident from the table values and from observing the graph of the function.

Step 1: Numerical Estimation

To complete the table, input the values \(10^{0}, 10^{1}, 10^{2}, 10^{3}, 10^{4}, 10^{5}, 10^{6}\) into the function \(f(x)=x^{2}-x \sqrt{x(x-1)}\). Compute the corresponding \(f(x)\) values for each.

Step 2: Graphical Estimation

Use a graphing tool to graph the function \(f(x)=x^{2}-x \sqrt{x(x-1)}\). Observe the end behaviour of the function as \(x\) approaches infinity. The limit as \(x\) approaches infinity is the y-value the graph approaches as we move to the right.

Step 3: Analytical Calculation

To find the limit analytically, apply the limit laws. Multiply the function by \(\frac{1}{x^2}\) then rearranging terms and simplify the expression. The limit as \(x\) approaches infinity of \(f(x)=x^{2}-x \sqrt{x(x-1)}\) will be the constant term of the simplified function.

Key Concepts

Graphing Utility

In calculus, a graphing utility is an essential tool that can help us understand the behavior of functions, especially as they approach infinity. By graphing a function, we can visually investigate where the curve tends to settle, which is crucial in estimating limits. For this exercise, to graph the function \( f(x) = x^2 - x \sqrt{x(x-1)} \), we need to use a graphing tool like a graphing calculator or a software application.

Here are simple steps you can follow with a graphing utility:

• Input the function \( f(x) = x^2 - x \sqrt{x(x-1)} \) into the graphing tool.
• Set the range for \( x \) from a small value to large values, such as \( 10^0 \) to \( 10^6 \).
• Observe the graph to see where the function appears to level out as \( x \) increases towards infinity.

The graph will generally show the function approaching a horizontal asymptote, indicating the limit of the function as \( x \) approaches infinity. Using a graph helps to visually estimate this limiting value and makes understanding calculus concepts an easier task.

Limit Analysis

Limit analysis involves studying the behavior of a function as the input approaches a particular value or infinity. In this exercise, the task is to estimate the limit of \( f(x) = x^2 - x \sqrt{x(x-1)} \) as \( x \) approaches infinity. Limits are fundamental in calculus as they describe how a function behaves at boundary conditions.

To perform limit analysis:

• Start by simplifying the function, if possible, to easier terms for evaluation.
• Consider multiplying by forms like \( \frac{1}{x^2} \) to simplify complexities in polynomial or radical expressions.
• Evaluate how each term in \( f(x) = x^2 - x \sqrt{x(x-1)} \) changes with very large values of \( x \).

By analyzing each term, the complexity reduces to focusing on terms that retain influence as \( x \) increases. Recognizing these behaviors analytically will reaffirm or adjust the graphical estimates, providing a precise limit value where applicable.

Function Behavior

Understanding function behavior is essential in determining how changes in \( x \) influence the entire function. For the function \( f(x) = x^2 - x \sqrt{x(x-1)} \), behavior as \( x \) approaches infinity gives insight into the long-term trends and limits.

Several factors to consider in function behavior:

• Identify the dominant term in the function that influences as \( x \) becomes very large.
• For \( f(x) = x^2 - x \sqrt{x(x-1)} \), observe if \( x^2 \) or the radical term has a lasting impact as \( x \) increases.
• Consider any horizontal or vertical asymptotes that might suggest a tendency toward a limit.

In general, if the highest power of \( x \) tends to cancel out or are dominated by coefficients acting as multipliers, it often simplifies understanding how the function behaves. Knowing these behaviors assists in predicting and confirming the limit of the function analytically as \( x \) grows indefinitely.