Limit of (1 + 1/x)^x as x approaches infinity

Published on October 19, 2024

Calculus

The limit of (1 + 1/x)^x as x approaches infinity is e.

Question

Find the limit of h(x) = (1 + 1/x)^x as x approaches infinity

Answer

Let’s find the limit of h(x) = (1 + 1/x)^x as x approaches infinity. This is a classic limit that defines the mathematical constant “e”. Here’s how we can approach it:

1. Recognize the Indeterminate Form

Direct substitution of x = ∞ leads to the indeterminate form 1^∞. We need to manipulate the expression to get a better understanding.

2. Utilize the Properties of Logarithms and Exponents

Introduce a Natural Logarithm: Let y = (1 + 1/x)^x. Taking the natural logarithm of both sides, we get:
Use Logarithm Property: Using the property ln(a^b) = b * ln(a), we can simplify:

3. Rewrite for L’Hôpital’s Rule

Our goal is to manipulate the expression so we can apply L’Hôpital’s Rule. Let’s rewrite the right side of the equation:

Now, as x approaches infinity, we have the indeterminate form 0/0, which allows us to apply L’Hôpital’s Rule.

4. Apply L’Hôpital’s Rule

L’Hôpital’s Rule states that if the limit of f(x)/g(x) as x approaches a certain value is an indeterminate form (like 0/0 or ∞/∞), then:

Let’s differentiate the numerator and denominator of our expression separately:

Numerator: The derivative of ln(1 + 1/x) with respect to x is: (1/(1 + 1/x)) * (-1/x^2) = -1/(x^2 + x)
Denominator: The derivative of 1/x with respect to x is -1/x^2.

Now we have: