%5Ex-e-(1%2B1%2Fx)%5Ex%2B1%2F2.jpeg "(1+1/x)^x E (1+1/x)^x+1/2")
Exploring the Limits of (1 + 1/x)^x and (1 + 1/x)^(x+1/2)
The expressions (1 + 1/x)^x and (1 + 1/x)^(x+1/2) are fascinating mathematical constructs that exhibit intriguing behavior as x approaches infinity. Let's delve into their properties and see how they relate to the famous mathematical constant e.
Understanding the Limit of (1 + 1/x)^x
As x approaches infinity, the expression (1 + 1/x)^x converges to a specific value. This value is known as the mathematical constant e, which is approximately 2.71828.
Here's why:
- The base: As x increases, the term (1 + 1/x) gets closer and closer to 1.
- The exponent: The exponent x grows infinitely large.
This seemingly contradictory situation – a base approaching 1 and an exponent approaching infinity – results in a finite and non-trivial limit.
Connecting (1 + 1/x)^(x+1/2) to e
The expression (1 + 1/x)^(x+1/2) can be rewritten as:
(1 + 1/x)^(x+1/2) = (1 + 1/x)^x * (1 + 1/x)^(1/2)
We already know that (1 + 1/x)^x approaches e as x approaches infinity. Now, let's examine the second part:
- (1 + 1/x)^(1/2) approaches 1 as x approaches infinity. This is because as x gets very large, 1/x becomes very small, and adding it to 1 has minimal impact.
Therefore, as x approaches infinity, (1 + 1/x)^(x+1/2) converges to e * 1 = e.
Key Points
- (1 + 1/x)^x and (1 + 1/x)^(x+1/2) both approach e as x approaches infinity.
- The limit of (1 + 1/x)^(x+1/2) is directly related to the limit of (1 + 1/x)^x through the manipulation of exponents.
This exploration highlights the profound connection between these seemingly complex expressions and the fundamental mathematical constant e, illustrating the beauty and power of limits in calculus.