%5Ex-limit.jpeg "(1-1/x)^x Limit")
Understanding the Limit of (1 - 1/x)^x
In the realm of calculus, limits play a crucial role in understanding the behavior of functions as their input approaches a certain value. One particularly intriguing limit is that of (1 - 1/x)^x as x approaches infinity. This limit holds significant importance as it directly relates to the mathematical constant 'e'.
The Intuition Behind the Limit
Before delving into the formal proof, let's grasp the intuition behind why this limit approaches 'e'. Consider the expression (1 - 1/x)^x. As x gets larger, the term 1/x becomes smaller. Consequently, the base of the expression, (1 - 1/x), approaches 1. Now, raising 1 to any power always results in 1. However, the power, x, is also growing infinitely large, creating a conflict.
This conflict is resolved by the subtle interplay between the base approaching 1 and the exponent growing infinitely large. It turns out that the limit converges to a specific value, which is the famous mathematical constant 'e'.
A Formal Approach to the Limit
To rigorously demonstrate that the limit of (1 - 1/x)^x as x approaches infinity equals 'e', we can use L'Hopital's rule. This rule allows us to evaluate limits of indeterminate forms such as 0/0 or ∞/∞.
-
Rewrite the expression: Let y = (1 - 1/x)^x. Taking the natural logarithm of both sides, we get ln(y) = x * ln(1 - 1/x).
-
Apply L'Hopital's rule: As x approaches infinity, ln(y) approaches the indeterminate form ∞ * 0. To apply L'Hopital's rule, we rewrite the expression as: ln(y) = ln(1 - 1/x) / (1/x). Now, as x approaches infinity, both the numerator and denominator approach 0.
-
Differentiate numerator and denominator: Taking the derivative of the numerator and denominator with respect to x, we get: (d/dx) ln(1 - 1/x) = 1/(x^2 - x) and (d/dx) (1/x) = -1/x^2.
-
Evaluate the limit: Applying L'Hopital's rule, we now have: lim (x->∞) ln(y) = lim (x->∞) [1/(x^2 - x)] / [-1/x^2] = -1.
-
Solve for y: Since ln(y) = -1, we have y = e^-1.
Therefore, we have shown that the limit of (1 - 1/x)^x as x approaches infinity equals e^-1 = 1/e.
Significance of the Limit
This limit holds significance in various fields, including:
- Compound Interest: The limit represents the maximum value of compounded interest when interest is compounded continuously.
- Probability: It arises in probability calculations related to Poisson processes.
- Calculus: It forms the basis for deriving the derivative of exponential functions.
In essence, understanding the limit of (1 - 1/x)^x unlocks a deeper understanding of the fundamental constant 'e' and its widespread applications in mathematics and various scientific disciplines.