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. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5} & 10^{6} \\ \hline f(x) & & & & & & & \\ \hline \end{array} $$ $$ f(x)=x^{2}-x \sqrt{x(x-1)} $$

Short 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-by-step solution

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.

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.

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.