Question

Use a spreadsheet to complete the table for the specified values of \(a\) and \(n\) to demonstrate that \(\lim _{x \rightarrow \infty} x^{n} e^{-a x}=0, \quad a>0, n>0\) \begin{tabular}{|l|l|l|l|l|} \hline\(x\) & 1 & 10 & 25 & 50 \\ \hline\(x^{n} e^{-a x}\) & & & & \\ \hline \end{tabular} $$ a=2, n=4 $$

Short answer

Upon calculation, for the given equations it can be observed that as \(x\) increases, \(x^n e^{-ax}\) does indeed approach zero. Thus verifying that \(\lim _{x \rightarrow \infty} x^{n} e^{-a x}=0\), for \(a>0\), \(n>0\).

Step-by-step solution

Setting up the spreadsheet

The first step is to set up the spreadsheet. Enter the \(x\) values, in increasing order, into the spreadsheet (1, 10, 25, and 50). In another column, label it as \(x^n e^{-ax}\). This column will be used to display the calculations of the function \(x^n e^{-ax}\) for each corresponding \(x\) value.

Calculation of function values

Now, compute the function \(x^n e^{-ax}\) for each entered \(x\) value. Since \(a = 2\) and \(n = 4\), the function becomes \(x^4 e^{-2x}\). For example, when \(x = 1\), the function value is \( 1^4 * e^{-2*1} \) and so on. Apply similar calculation for the rest of the \(x\) values.

Observing the trend

Once all the calculations have been carried out, take note of the trend of the results. As \(x\) tends to infinity, does \(x^n e^{-ax}\) approach zero? This shall confirm or disprove the given limit problem statement.

Key concepts

Spreadsheet Calculations

Let's dive into how spreadsheets can simplify complex mathematical tasks, like calculating the limit of an exponential function. First, spreadsheets like Excel or Google Sheets help in organizing data neatly. They also allow for automatic calculations, which reduces human error.

To start, you simply enter the values of interest, which are the given fixed values for the variable \(x\) in our exercise. These are entered in a straightforward column format. The beauty of spreadsheets lies in their ability to compute large sets of data quickly. By inputting the formula \(x^4 e^{-2x}\) into the spreadsheet for each \(x\) value, spreadsheets generate results without the need to compute each value manually.

In this case, you create a formula within the spreadsheet that calculates the product of \(x^4\) multiplied by the exponential function \(e^{-2x}\) for each \(x\), populating the values automatically. This ease of replication and accuracy proves invaluable, especially when correcting or exploring different values in your function.

Exponential Decay

Exponential decay is an essential concept in mathematics, particularly in modeling scenarios where quantities reduce over time. In our example, the term \(e^{-ax}\) represents exponential decay.

The function \(x^n e^{-ax}\) incorporates exponential decay through \(e^{-ax}\), as \(a > 0\) ensures the decay as \(x\) increases. This means that while the polynomial component \(x^4\) grows, the exponential component \(e^{-2x}\) shrinks rapidly enough that the entire expression \(x^4 e^{-2x}\) tends to zero for very large \(x\).

Exponential decay in this context supercedes the polynomial growth rate of the \(x^4\) term, proving that the limit of \(x^n e^{-ax}\) approaches zero as \(x\) approaches infinity. This phenomenon illustrates how even a rapidly growing polynomial can be overpowered by an exponential decay when exploring behavior at infinity.

Polynomial Growth

Polynomial growth refers to the rate at which a polynomial expression increases. In our exercise, this is represented by the term \(x^n\), specifically \(x^4\) given \(n = 4\).

The polynomial component \(x^4\) signifies that as \(x\) grows, \(x^4\) increases very quickly. Often, polynomial growth is straightforward and predictable, with a clear pattern of escalation. However, in our function, this growth is contrasted with the exponential decay rate.

While \(x^4\) alone would continue to grow without bound, the expression \(x^4 e^{-2x}\) shows a powerful interplay between growth and decay, demonstrating how exponential decay eventually dominates, leading to an overall decline in function value. Understanding this balance is crucial in evaluating limits of functions involving both growth and decay components.