Problem 79 - College Algebra Essentials

Question

Use a calculator to evaluate \(\left(1+\frac{1}{x}\right)^{x}\) for \(x=10,100,1000\) \(10,000,100,000,\) and \(1,000,000 .\) Describe what happens to the expression as \(x\) increases

Step-by-Step Solution

Answer

As \(x\) increases, the expression \(\left(1+\frac{1}{x}\right)^{x}\) appears to be converging to a constant value which is approximately equal to the Euler's number, \(e\), approximately 2.71828. This hints at the principle in calculus that the expression \(\lim_{{x \to \infty}} \left(1+\frac{1}{x}\right)^{x} = e\).

Step 1: Plug in x = 10

Use a calculator to find the value of \(\left(1+\frac{1}{10}\right)^{10}\) using the rules of exponents and arithmetic.

Step 2: Plug in x = 100

Next, use a calculator to compute the value of \(\left(1+\frac{1}{100}\right)^{100}\). Observe if there's a trend starting to form.

Step 3: Repeat for further values

Continue to plug in the given values for \(x\): 1000, 10,000, 100,000, and 1,000,000 and calculate \(\left(1+\frac{1}{x}\right)^{x}\) for each. Pay attention to the trend, how the calculated value is changing as \(x\) increases.

Step 4: Analyze the trend

Upon calculating all values, look closely at the results. What happens to the expression \(\left(1+\frac{1}{x}\right)^{x}\) as \(x\) increases? Is there a trend to be observed?

Key Concepts

Exponential Growth

Exponential growth is a powerful concept where a quantity increases repeatedly by a consistent and significant rate. When dealing with exponential expressions such as \(\left(1+\frac{1}{x}\right)^{x}\), we see how the base of the exponent contains a fraction that decreases as \(x\) increases. Despite the decreasing fraction, the expression itself reaches extraordinary outcomes due to the properties of exponentials.

As you compute \(\left(1+\frac{1}{x}\right)^{x}\) for increasing values of \(x\), you will notice this function approaching a special number. Initially, the exponential behavior is observed through increasing growth. However, the speed of this growth slows down. This phenomenon demonstrates exponential stabilization rather than continuous surge. Ultimately, the outcome reveals the boundary at which exponential growth converges—an important concept when exploring the behavior of advanced mathematical functions.

Numerical Analysis

Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. It's a useful tool when exact results are difficult to achieve. In the exercise with \(\left(1+\frac{1}{x}\right)^{x}\), calculators are employed within this concept to approximate values as \(x\) dramatically increases.

Using your calculator, you can plug in the following values: 10, 100, 1,000, and beyond. Numerical analysis in this case helps one observe a pattern. The approach gives a practical demonstration that allows examination of the values calculated and how they approach a specific limit, which cannot always be easily derived by pure algebraic manipulation.

This method emphasizes the importance of numerical solutions and simulations in real-life applications where exact forms of a solution might not be feasible. It is especially useful in engineering contexts and situations involving complex systems, where approximations still lead to insightful predictions and efficient solutions.

Convergence

Convergence is when a sequence or function approaches a specific value as its index or input grows. In the given exercise, the expression \(\left(1+\frac{1}{x}\right)^{x}\) provides a valuable opportunity to witness convergence.

Upon calculating values for increasingly large \(x\), it becomes apparent that this expression approaches a limit. The magic number here is \(e\), a fundamental constant approximately equal to 2.71828. As \(x\) grows indefinitely, \(\left(1+\frac{1}{x}\right)^{x}\) does not keep growing exponentially but instead converges on \(e\).

Recognizing this type of convergence shines a light on the broader mathematical principle of limits. It is crucial for understanding not only static outcomes but also dynamic and iterative processes in calculus and applied mathematics.