Problem 47 - Precalculus with Limits

Question

In Exercises 37-48, use a graphing utility to graph the function and approximate the limit accurate to three decimal places. $$\lim_{x \to 0} (1-x)^{2/x}$$

Step-by-Step Solution

Answer

Under ideal conditions, the function will approach the value of Euler's number \(e\) when approximated to three decimal places. However, depending on the graphing utility in use, the value can be slightly off but must be accurate to three decimal places.

Step 1: Interpret the Function

The given function is \((1-x)^{2/x}\). Here, \(x\) is approaching 0. So, the basic task is to find what value the function approaches as \(x\) tends to 0.

Step 2: Graph the function using a Graphing Utility

Use a graphing utility like Desmos, GeoGebra, or a scientific calculator that can graph this function. After graphing, it can be seen that the curve of the function approaches a certain value when \(x\) tends to 0. It's important to find the value as accurately as possible.

Step 3: Approximate the limit

By observing or computing numerically on the graphing utility, approximate the value of the function when \(x\) is close to 0. Ensure the approximation is up to three decimal places.

Key Concepts

Graphing Utilities

Graphing utilities are powerful tools that help visualize mathematical functions. They include software like Desmos, GeoGebra, and scientific calculators capable of plotting functions. These utilities allow users to input the mathematical expression, such as \((1-x)^{2/x}\), and instantly see a graph of how it behaves.Understanding how to use these tools effectively is crucial:

• You can zoom in or out on particular parts of the graph.
• These tools let you input specific values for \(x\) to see the corresponding \(y\)-values.
• They often provide numerical approximation features, helping you find the value to which the function approaches.

Using graphing utilities concretely brings to life abstract math concepts and enhances comprehension by offering visual representations.

Approximate Limits

In calculus, limits express the value that a function approaches as the input approaches a particular point. In our example, the function \(\lim_{x \to 0} (1-x)^{2/x}\) approaches a specific limit as \(x\) tends towards zero.To approximate limits:

• Observe how the function behaves graphically around the point of interest, in this case, as \(x\) gets closer to 0.
• Use the graphing utility's feature to evaluate the function at values of \(x\) that are incrementally close to the point.
• Identify any patterns or asymptotic behavior that indicate the function's limiting value.

In this problem, plotting the graph allows one to see that as \(x\) approaches 0, the function approaches a value that can be recorded up to three decimal places, providing a numerical estimate of this behavior.

Exponential Functions

Exponential functions, like \((1-x)^{2/x}\), refer to expressions where variables appear in the exponent. Though this specific example looks somewhat different due to its form, it still exhibits exponential characteristics.Key traits of exponential functions include:

• Rapid growth or decay depending on the function's base and exponent.
• In continuous models, they describe a wide range of naturally occurring phenomena, such as population growth and radioactive decay.
• Desirable mathematical properties like always being continuous and differentiable.

In the given problem, the function evolves rapidly as \(x\) approaches 0, displaying subtle exponential nature by its ability to approach a limit quickly. Understanding the foundational rules of exponential growth and decay aids in predicting and confirming the behavior seen through graphing utilities, connecting theoretical understanding with practical approximation.