Problem 83 - Precalculus

Question

Plot the function \(y=\left(1+\frac{1}{x}\right)^{x} .\) What is the horizontal asymptote as \(x\) increases?

Step-by-Step Solution

Answer

The horizontal asymptote as \(x\) increases is \(y = e\).

Step 1: Understand the Function

The function given is \( y = \left(1+\frac{1}{x}\right)^x \). This function is a common representation of the exponential limit definition of the mathematical constant \( e \). As \( x \) becomes very large, the expression approaches \( e \).

Step 2: Determine the Behavior at Infinity

The expression \( \left(1+\frac{1}{x}\right)^x \) is known to converge to \( e \) as \( x \) approaches infinity. This property relies on the definition of the number \( e \), where:\[e^{x} =\lim_{{x \to \infty}} \left(1 + \frac{1}{x}\right)^x = e \]

Step 3: Identify the Horizontal Asymptote

Since the function approaches the constant \( e \) as \( x \) increases towards infinity, the horizontal asymptote of the graph is at \( y = e \). This line represents the value that the function approaches but does not actually reach as \( x \) becomes very large.

Key Concepts

Exponential Function

An exponential function is a type of mathematical function where a constant base is raised to a variable exponent. This is often written in the form of \(a^x\), where \( a \) is a positive constant and \( x \) is the variable exponent. These functions are important in many areas of mathematics and science because they model growth or decay processes, such as population growth or radioactive decay.
An interesting property of exponential functions is their rate of change, which increases (or decreases) at a rate proportional to their current value. Thus, exponential functions can exhibit rapid increases or declines under certain conditions.

• For positive growth, the base of the exponential function \( a \) is greater than 1, leading to an increase in the value of the function as \( x \) increases.
• For decay, the base is between 0 and 1, causing the function to get closer to zero as \( x \) increases.

Exponential functions also have a horizontal asymptote, which is a line that the graph of the function approaches but never quite reaches. This is important for understanding long-term behavior as the input value becomes very large or very small.

Limit of a Function

The limit of a function is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. In simple terms, it tells us what value a function is getting closer to, but not necessarily reaching. In mathematical notation, the limit of a function \(f(x)\) as \(x\) approaches \(a\) is expressed as \(\lim_{x \to a} f(x)\).
Limits are crucial when one wants to understand how functions behave over the long term, like in our original exercise, to find the horizontal asymptote of the function.

• When finding the limit at infinity, we observe the behavior of the function as \( x \) tends to positive or negative infinity.
• If a function approaches a finite number as \( x \) tends towards infinity or negative infinity, this number is the horizontal asymptote.

Calculating limits allows us to predict outcomes and understand the constraints of different functions, which is essential for fields like calculus and real-world problem-solving.

Mathematical Constant e

The mathematical constant \( e \) is one of the most important numbers in mathematics, approximately equal to 2.71828. It is the base of natural logarithms and frequently appears in scenarios involving growth processes. In the context of our exercise, \( e \) is the limit of \( \left(1 + \frac{1}{x}\right)^x \) as \( x \) approaches infinity. Hence, this demonstrates its vital role in exponential functions and calculus.
\( e \) arises naturally in many situations where systems involve continuous growth or decay.

• This constant is crucial in the study of exponential growth, such as populations or investments.
• \( e \) is defined as the unique real number such that the derivative of the function \( e^x \) is equal to \( e^x \) itself, making it pivotal in calculus.

Understanding \( e \) enhances comprehension of more complex mathematics and aids in analyzing endless growth processes effectively.