Problem 79 - College Algebra

Question

Use a calculator to evaluate \(\left(1+\frac{1}{x}\right)^{x}\) for \(x-10,100,1000\) \(10,000,100,000,\) and \(1,000,009 .\) Describe what happens to the expression as \(x\) increases

Step-by-Step Solution

Answer

The expression \(\left(1+\frac{1}{x}\right)^{x}\) approaches the number \(e ≈ 2.71828\) as \(x\) increases.

Step 1: Substitute the first number for x

Substitute \(x = 10\) into the expression: \((1+\frac{1}{10})^{10}\)

Step 2: Substitute the second number for x

Substitute \(x = 100\) into the expression: \((1+\frac{1}{100})^{100}\)

Step 3: Continue the substitutions

Perform these substitutions for all the rest of the values, \(x = 1000, 10000, 100000, 1000009\), respectively, observing how the results are affected with each ensuing substitution.

Step 4: Discover the pattern and conclude

Observe the pattern that emerges from the results of the substitutions. When conducted the calculations will all result in an approximation of a specific mathematical constant, namely Euler's number \(e ≈ 2.71828\). Conclude that this pattern continues for higher and higher values of \(x\) and that the expression \(\left(1+\frac{1}{x}\right)^{x}\) approaches \(e\) as \(x\) increases.

Key Concepts

Limit of a Sequence

Understanding the concept of the limit of a sequence is crucial in calculus. It helps us grasp how sequences behave as their terms extend towards infinity. A sequence is essentially a list of numbers arranged in a specific order. For example, consider the sequence defined by the formula:

• \(a_n = \left(1+\frac{1}{n}\right)^{n}\)

Now, as \(n\) becomes larger, the values of \(a_n\) start to settle at or "approach" a number, which is known as the limit of the sequence.
If a sequence approaches a definite number as the terms increase indefinitely, it is said to converge, and that number is the sequence's limit. In our example, as \(x\), or \(n\), increases to a very large number, \(\left(1+\frac{1}{x}\right)^{x}\) approaches the number \(e\). This behavior is a classic illustration of convergence and helps students appreciate the elegance of sequences in mathematical analysis.

Euler's Number

Euler's number, often represented as \(e\), is an important irrational constant in mathematics, approximately equal to 2.71828. It frequently appears in various fields such as finance, biology, and physics due to its unique properties.
One fascinating aspect of \(e\) is revealed through the expression \(\left(1+\frac{1}{n}\right)^{n}\). As we input larger values for \(n\), the expression approximates \(e\). This is particularly interesting because it shows how a simple repetitive pattern links to a fundamental constant.
Euler's number is integral to the concept of natural growth. Functions that involve \(e\) beautifully describe how quantities evolve continuously over time. When working with limits and exponential growth, \(e\) becomes not just a number, but a powerful tool connecting theory with practical applications.

Approaching a Limit

The idea of approaching a limit is about observing what value a function or sequence appears to be getting closer to, as the input or index approaches some value. In calculus, seeing how functions behave as they approach certain inputs offers insight into the overall nature of mathematical behaviors.
For example, in the sequence \(\left(1+\frac{1}{x}\right)^{x}\), as \(x\) approaches infinity, the calculated values get closer to \(e\). This phenomenon is a core concept of limits, showing how functions and sequences converge to specific values as the inputs grow large or approach a particular point.

• Approaching a limit involves gaining precision through larger or smaller input values.
• It showcases the behavior of functions or sequences very precisely.

Limits are foundational in calculus because they lead to defining derivatives and integrals, governing a lot of mathematical analysis. These concepts make it easier to understand how different mathematical processes interact with each other over timescales or through changes in variables.