Problem 32 - Calculus

Question

Estimate the limit of \(\left(1+\frac{1}{x}\right)^{x}\) as \(x \rightarrow \infty\).

Step-by-Step Solution

Answer

The limit is \(e\).

Step 1: Recognize the Formula of Interest

The limit we need to evaluate is \[\lim_{{x \to \infty}} \left( 1 + \frac{1}{x} \right)^{x}.\]This expression resembles a form that is commonly known to approach the mathematical constant \(e\), which is approximately equal to 2.71828.

Step 2: Understanding the Limit Definition of e

The expression \[\left( 1 + \frac{1}{x} \right)^{x}\]is equivalent to the definition of the mathematical constant \(e\) as follows:\[e = \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^{n}.\]By this definition, as \(x\) increases towards infinity, our expression will approach \(e\).

Step 3: Evaluate the Limit

Using the known limit definition of \(e\), we directly evaluate the limit:\[\lim_{{x \to \infty}} \left( 1 + \frac{1}{x} \right)^{x} = e.\]Hence, the expression approaches \(e\) as \(x\) tends toward infinity.

Key Concepts

Mathematical Constant e

The mathematical constant \( e \) is an essential number in mathematics that often appears in calculus, particularly when dealing with growth or decay processes. It is approximately equal to 2.71828 and is sometimes called Euler's number. This constant is incredibly important because it serves as the base for natural logarithms.
To understand \( e \), envision a scenario where you're earning compound interest. If you make investments where your interest is reinvested rapidly — let's say ‛continuously’ — over time, the total accumulated amount will grow closer to \( e \), given certain conditions. This idea of continuous growth directly ties into how \( e \) is defined mathematically.

• \( e \) is the limit of \( \left( 1 + \frac{1}{n} \right)^n \) as \( n \) approaches infinity.
• It symbolizes the idea of maximizing continuous growth or decay.
• Numerous natural and physical phenomena model their behavior using \( e \).

Limit Definition

The concept of a limit is fundamental in calculus. It is the value that a function or sequence "approaches" as the input or index approaches some value. Limits are critical for defining derivatives and integrals.
In our exercise, the function is \( \left( 1 + \frac{1}{x} \right)^{x} \) and we are looking at what it approaches as \( x \) goes to infinity. According to the definition of \( e \), as \( x \) grows larger and larger, this expression becomes closer to \( e \).

• Limits help find the behavior of functions as they approach a specific input value.
• They give precise meaning to concepts like continuity, derivatives, and integrals.
• The limit process can be used to determine important constants like \( e \).

Understanding limits allows for better insights into the subtler behaviors of functions and sequences in mathematics.

Infinite Limits

When we discuss infinite limits, we're exploring what happens to a function as its input becomes infinitely large. It helps in understanding how certain mathematical expressions behave as they extend indefinitely.
In our case, we examined how \( \left( 1 + \frac{1}{x} \right)^{x} \) behaves as \( x \) approaches infinity. Its limit is the constant \( e \). The concept of infinite limits often appears in both calculus and real-world phenomena where variables take on unbounded values.

• Infinite limits deal with the behavior of functions as the input moves towards positive or negative infinity.
• They're crucial for defining end behavior in both calculus and real-world applications.
• Understanding such limits ensures a comprehension of how various mathematical expressions behave in the long term.

Mastering infinite limits allows students and mathematicians to predict and understand the long-term behavior of sequences and functions.