Problem 66 - Introductory and Intermediate Algebra for College Students 4th

Question

Use a calculator to obtain an approximate value for \(e\) to as many decimal places as the display permits. Then use the 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

After finding that the approximate value of \(e\) is 2.71828, evaluating the expression \(\left(1+\frac{1}{x}\right)^{x}\) for increasing values of \(x\) shows that the result approximates to \(e\). As \(x\) increases, the results get closer to the value of \(e\).

Step 1: Compute the Value of \(e\)

Using a calculator with a capability for calculating the exponential function, input the value of \(e\). This should provide an estimate of \(e\) which is approximately 2.71828.

Step 2: Compute the Expression for Various \(x\)

Next, compute the expression \(\left(1+\frac{1}{x}\right)^{x}\) for \(x = 10, 100, 1000, 10000, 100000\) and \(1000000\). Observe how the result changes as \(x\) increases.

Step 3: Analyze the Results

As \(x\) increases, it will be observed that the result of the expression approximates closer and closer to the value of \(e\). This is because \(\left(1+\frac{1}{n}\right)^n\) is the limit definition of the exponent \(e\).

Key Concepts

Exponential Functions

Exponential functions are a foundational part of algebra and calculus, describing situations where a quantity grows or decays at a rate proportional to its current value. Mathematically, they are expressed as functions of the form:

• \( y = a^x \)

where \( a \) is a constant base and \( x \) is the exponent.
Exponential functions are important because they model a wide variety of phenomena in nature, economics, and technology, such as population growth, radioactive decay, and compound interest.
In the context of this exercise, the expression \( \left(1 + \frac{1}{x}\right)^x \) is an exponential expression that, as \( x \) becomes very large, approaches a special number, noted as \( e \). Understanding how this approximation occurs is crucial for grasping the behavior of exponential functions in more complex scenarios.

Limit Definition

In mathematics, the concept of a limit is used to describe the behavior of a function as its input approaches a particular value or infinity.
In this exercise, we explore the limit definition specific to finding Euler's number, \( e \). The expression \( \left(1+\frac{1}{x}\right)^x \) is defined as the limit of the function as \( x \) tends to infinity:

• \[ e = \lim_{{x \to \infty}} \left(1+\frac{1}{x}\right)^x \]

This shows how \( e \) emerges from repeatedly multiplying a base adjusted by an inverse relation to \( x \), and progressively increasing \( x \).
Using limits helps in understanding how sequences and functions converge to a certain point. In this exercise, what we observe is that as \( x \) increases, the sequence of computed values gets closer to \( e \), demonstrating the convergence that is quintessential to the study of limits.

Euler's Number

Euler's Number, denoted as \( e \), is a fundamental mathematical constant approximately equal to 2.71828.
It is critical in various branches of mathematics, due to its unique properties in calculus, particularly as a base for natural logarithms and in describing growth processes that are continuous rather than step-based.
As the exercise illustrates, Euler's Number is what the expression \( \left(1+\frac{1}{x}\right)^x \) reaches as \( x \) becomes exceedingly large.

• This property makes \( e \) the base of natural logarithms, because it is the only number such that the function \( f(x) = e^x \) has a derivative equal to \( f(x) \) itself at every point.
• It is also the backbone of the exponential function, which appears in many natural and theoretical exponential processes.

Understanding \( e \), its derivation, and its applications is essential for advanced calculations in continuous growth and decay, making it a pervasive element in mathematical analysis and beyond.