Part 4/5:
\lim_{x \to 0} \frac{1+x}{x} = e
]
This forms the core proof that ( e ) is indeed approximately equal to 2.718.
The Nature of E: Irrational and Transcendental
It’s essential to recognize that ( e ) is not just an ordinary number; it is classified as both an irrational and transcendental number. This means that ( e ) cannot be expressed as a simple fraction and does not resolve into a repeating decimal. Instead, it continues indefinitely without repeating, much like ( \pi ).
Calculating E
To get a tangible grasp of ( e )'s value, we can utilize calculators to approximate it. For small values, say ( x = 0.001 ), we can illustrate:
[
1 + \frac{0.001}{1} \approx e
]