Problem 49 - Calculus of a Single Variable

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

Step-by-Step Solution

Answer

The limit of the function as \(x\) approaches infinity is infinity.

Step 1: Numerical Estimation

Use a graphing utility to plug in the values for \(x\) and compute \(f(x)\). Typically, as \(x\) approaches larger values (like \(10^1\), \(10^2\), \(10^3\), ... , \(10^6\)), \(f(x)\) should also start to converge towards a certain limit.

Step 2: Graphical Estimation

Plot the function \(f(x) = x - \sqrt{x(x-1)}\) in a graphing utility. The \(y\)-value that our function gets closer to as \(x\) moves towards positive infinity will be the graphical estimate of our limit.

Step 3: Analytical Derivation

Calculate the limit of the function as \(x\) goes to infinity. As \(x\) grows large, the term \(- \sqrt{x(x-1)}\) in our function will have ever lesser impact on the value of \(f(x)\), making \(f(x)\) more and more similar to the simpler function \(x\). So the limit of \(f(x)\) as \(x\) approaches infinity will coincide with the limit of \(x\) as \(x\) approaches infinity, and that limit is infinity.

Key Concepts

Numerical Estimation

In order to numerically estimate the limit of the given function as \( x \) approaches infinity, we will assess the behavior of \( f(x) = x - \sqrt{x(x-1)} \) for very large values of \( x \). Essentially, numerical estimation involves calculating \( f(x) \) for sequentially larger values of \( x \), capturing a trend. To do this effectively, use a graphing utility to substitute values such as \( x = 10^0, 10^1, 10^2, 10^3, 10^4, 10^5, \) and \( 10^6 \). For each computation:

• Substitute the \( x \) value into the equation.
• Calculate \( \sqrt{x(x-1)} \) and subtract it from \( x \).
• Record the result for each \( x \).

As you perform these calculations, you will notice that \( f(x) \) tends to stabilize or approach a pattern as \( x \) increases, which is the basis for estimating the limit numerically.

Graphical Estimation

Graphical estimation of a limit involves visually identifying the behavior of the function \( f(x) = x - \sqrt{x(x-1)} \) as \( x \) increases, by plotting it on a graph. To achieve this, follow these steps:

• Use a graphing utility, like a calculator or software, to graph the function over an appropriate set of \( x \) values, extending into the range of very large numbers, such as \( 10^0 \) to \( 10^6 \).
• Observe the graph; specifically, note the \( y \)-value at which \( f(x) \) appears to stabilize.
• This stable \( y \)-value as \( x \) progresses towards infinity is your visual clue to the estimated limit of the function.

Plotting functions provides insight into their long-term behavior and allows for observational validation that complements other estimation methods.

Analytical Derivation

Analytical derivation of limits requires evaluating the function symbolically and logically as \( x \) approaches infinity. Given \( f(x) = x - \sqrt{x(x-1)} \), notice the form of the expression.The term \( \sqrt{x(x-1)} \) expands to \( \sqrt{x^2 - x} \). As \( x \) becomes very large, \( x^2 \) dominates, making \( \sqrt{x(x-1)} \) closely resemble \( \sqrt{x^2} = x \). Therefore, the expression for the function simplifies:\[f(x) = x - \sqrt{x(x-1)} \approx x - x = 0\]However, since the next leading term \( \frac{-x}{2x} \) (originating from \( \sqrt{x(x-1)} \approx x - \frac{1}{2} \)) refines the approximation, we understand that \( f(x) \) actually converges to a value approaching \( 1/2 \) instead of \( 0 \).So, analytically, the limit of \( f(x) \) as \( x \) approaches infinity is determined as \( 1/2 \), affirming the numerical and graphical estimations previously concluded.