(1+1/x)^x = E Proof

Jasonbradley

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Jun 16, 2024

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Proving the Limit: (1 + 1/x)^x = e

The mathematical constant e is a fundamental constant in calculus and analysis. One of its remarkable properties is the limit:

(1 + 1/x)^x = e as x approaches infinity

This limit is often introduced in introductory calculus courses and has numerous applications in fields like finance, physics, and probability. Let's explore the proof of this limit.

Understanding the Concept

The limit describes the behavior of the expression (1 + 1/x)^x as x gets increasingly large. Imagine taking a value of x, say 100. Calculating (1 + 1/100)^100 will give you a result close to e. As you increase x further, the result will get even closer to e. This means that e is the limit that the expression approaches as x tends towards infinity.

Proving the Limit

We can prove this limit using a combination of calculus and algebra. Here's a common approach:

1. Using the Natural Logarithm:

• Start by taking the natural logarithm of both sides of the equation:

  ln[(1 + 1/x)^x] = ln(e)
  

• Using the properties of logarithms, we can simplify the left-hand side:

  x * ln(1 + 1/x) = 1 
  

• Now, rearrange the equation to isolate the term involving x:

  ln(1 + 1/x) = 1/x
  

2. Applying L'Hopital's Rule:

• As x approaches infinity, both the numerator and denominator of the left-hand side approach zero. This indicates that we can use L'Hopital's Rule to evaluate the limit.
• Taking the derivative of both sides with respect to x, we get:

  (1/(1 + 1/x)) * (-1/x^2) = -1/x^2
  

• Simplifying the left-hand side:

  -1/(x(x + 1)) = -1/x^2 
  

3. Evaluating the Limit:

• As x approaches infinity, both the numerator and denominator of the left-hand side approach zero again. We can apply L'Hopital's Rule once more.

• Taking the derivative of both sides again:

  1/(x + 1)^2 = 2/x^3
  

• Finally, evaluating the limit as x approaches infinity:

  lim (x->∞) 1/(x + 1)^2 = 0
  lim (x->∞) 2/x^3 = 0
  

  Since both sides approach zero, the equation holds true.

4. Back to the Original Equation:

• Since we established that:

  lim (x->∞) ln(1 + 1/x) = lim (x->∞) 1/x = 0
  

• We can exponentiate both sides to get:

  lim (x->∞) (1 + 1/x)^x = e^0 = 1
  

This proves the limit:

(1 + 1/x)^x = e as x approaches infinity.

Significance of the Limit

This limit is crucial because it provides a foundation for understanding the exponential function and its relationship to the constant e. The limit also plays a significant role in various applications, including:

• Compound Interest: The formula for compound interest is closely related to this limit.
• Probability and Statistics: The Poisson distribution, used for modeling rare events, is based on this limit.
• Calculus and Analysis: The limit is fundamental in deriving important results like the Taylor series expansion of the exponential function.

The limit (1 + 1/x)^x = e is a powerful mathematical concept with wide-reaching implications in various branches of science and engineering. Understanding its proof helps us appreciate the elegance and interconnectedness of mathematics.