Problem 3 - Calculus of a Single Variable

Question

Numerical and Graphical Analysis In Exercises \(1-4\) , complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to support your result. $$ \lim _{x \rightarrow \infty} x^{5} e^{-x / 100} $$ $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {1} & {10} & {10^{2}} & {10^{3}} & {10^{4}} & {10^{5}} \\ \hline f(x) & {} & {} & {} & {} \\ \hline\end{array} $$

Step-by-Step Solution

Answer

The estimated limit of the function \(x^{5}e^{-x / 100}\) as \(x\) approaches infinity is 0.

Step 1: Fill the Table with Function Values

First, let's find the function values for the given \(x\) values. Substitute each \(x\) value into the function \(x^{5}e^{-x/100}\). \n\For \(x=1\), \(f(x)=f(1)=1^{5}e^{-1/100}= e^{-0.01}\). \n\For \(x=10\), \(f(x)=f(10)=10^{5}e^{-10/100}= 100000e^{-0.1}\). \n\For \(x=10^{2}\), \(f(x)=f(10^{2})=(10^{2})^{5}e^{-(10^{2})/100} =10^{10}e^{-1}\). \n\For \(x=10^{3}\), \(f(x)=f(10^{3})=(10^{3})^{5}e^{-(10^{3})/100} =10^{15}e^{-10}\). \n\For \(x=10^{4}\), \(f(x)=f(10^{4})=(10^{4})^{5}e^{-(10^{4})/100}= 10^{20}e^{-100}\). \n\For \(x=10^{5}\), \(f(x)=f(10^{5})=(10^{5})^{5}e^{-(10^{5})/100}=10^{25}e^{-1000}\). \n\Now the table becomes: \n\\[\begin{array}{|c|c|c|c|c|c|c|}\hline x & {1} & {10} & {10^{2}} & {10^{3}} & {10^{4}} & {10^{5}} \ \hline f(x) & {e^{-0.01}} & {100000e^{-0.1}} & {10^{10}e^{-1}} & {10^{15}e^{-10}} & {10^{20}e^{-100}} & {10^{25}e^{-1000}} \ \hline\end{array}\]

Step 2: Estimate the Limit

Observe the table: as \(x\) increases, the function value \(f(x)\) tends to decrease due to the exponential term \(e^{-x/100}\). When \(x\) becomes very large, the exponential term diminishes towards zero, which makes the overall function \(f(x) = x^{5}e^{-x/100}\) tends to zero as well. Therefore, we can estimate that the limit of the function as \(x\) approaches infinity is zero. That is, \(\lim_{x \rightarrow \infty} x^{5}e^{-x / 100} = 0.\)

Step 3: Graph and Check the Result

Finally, corroborate the result by graphing the function using a graphing utility. As \(x\) values get larger and larger, the function heads toward zero, which supports the numeric estimation.

Key Concepts

Limits at Infinity

Calculating limits at infinity involves understanding the behavior of a function as the input, or x-value, grows without bound. In other words, we want to know what value the function approaches as x becomes very large. For the function \( f(x) = x^5e^{-x/100} \), we observe that as \( x \to \infty \), \( e^{-x/100} \), an exponential decay factor, falls toward zero much faster than \( x^5 \) grows. Consequently, the function's value trends towards zero. This insight lets us estimate the limit of \( f(x) \) as \( x \to \infty \) to be zero, demonstrating an essential principle in calculus—dominance of exponential decay over polynomial growth at infinity.

Understanding how different functions behave as x goes to infinity is crucial in calculus. With polynomial functions, higher degrees tend to dominate the behavior as x grows larger. However, when coupled with an exponential function that decays, the exponential function typically dictates the long-term behavior, overpowering even high-degree polynomial terms.

Exponential Functions

Exponential functions, like \( e^{-x/100} \) in our exercise, play a critical role in various mathematical applications. They are characterized by a constant raised to a variable exponent. In \( y = e^x \), the base is Euler's number (\( e \) ≈ 2.71828), which is a fundamental constant in mathematics. When the exponent is negative, as in \( e^{-x} \) or our example \( e^{-x/100} \) the function represents exponential decay. This type of function decreases rapidly as x increases, and as such, they are integral in modeling processes with rapid decrease or decay, such as cooling temperatures or radioactive decay.

For students, the key to understanding exponential functions is recognizing their rapid growth or decay rate, which is why they often prevail in the limit at infinity context, overshadowing other terms in the function as they approach zero much faster than any polynomial term can grow.

Graphical Analysis

A picture can be worth a thousand words, especially in calculus. Graphical analysis involves plotting a function to visually understand its behavior. For the function \( f(x) = x^5e^{-x/100} \) in our problem, graphing is a powerful tool to corroborate our numerical findings. Upon visual inspection, we would see that for small values of x, the function rises sharply due to the \( x^5 \) term. But as x continues to grow, \( e^{-x/100} \) kicks in, pulling the values of \( f(x) \) down towards zero, creating a peak before heading towards the horizontal axis.

Using technology like graphing calculators or computer software, students can easily visualize functions and verify their results. This graphical approach adds another layer of understanding and confirmation to the analytical and numerical methods used to estimate limits.

Numerical Analysis

Numerical analysis is the study of algorithms that use numerical approximation for solving mathematical problems. When it comes to estimating limits, constructing a table of values, as done in our solution, is a simple yet effective numerical method. It allows us to see a pattern in how the function values change as x increases. In our exercise, we've calculated a small set of values for \( f(x) = x^5e^{-x/100} \) at specific increasing points of x. This can give us a rough approximation of where the function's value is heading.

Though computer software can handle more complex calculations and provide more precise numerical approximations, this method remains a fundamental and accessible tool to introduce students to the concept of estimating limits. It's a hands-on approach that reinforces understanding of function behavior.

Calculus

Calculus is a field of mathematics that studies change and motion through two primary branches, differential and integral calculus. It forms the basis for many scientific and engineering disciplines. In the context of our exercise, we use the concept of limits, a foundational calculus topic, to determine the behavior of \( f(x) = x^5e^{-x/100} \) as x reaches infinity. Calculus provides tools not only for evaluating such limits but also for understanding instantaneous rates of change (derivatives) and accumulation of quantities (integrals).

When approaching calculus problems, it's essential to apply a range of techniques, including numerical, analytical, and graphical methods, to develop a well-rounded understanding of the concepts at hand. The limit estimating exercise showcases how these parts of calculus come together to solve complex mathematical questions.