Limit definition of e

The number e is defined as the limit of a sequence in which 1 is added to an ever smaller quantity and the result is raised to an ever larger power. \[ \lim_{x \to +\infty} \left(1 + \frac{1}{x}\right)^x = e \] The constant e is an irrational number with approximate value 2.71828.

A distinctive feature of this limit is that, although it takes the indeterminate form $ 1^{\infty} $, it converges to a well-defined finite value.

This fundamental limit underpins many important formulas and standard limits. It defines one of the central constants in mathematics ( $ e = 2.71828... $ ) and lies at the heart of the exponential function $ e^x $, uniquely characterized by the property that its derivative coincides with the function itself.

Proof

Consider the limit

$$ \lim_{x \to +\infty} \left(1 + \frac{1}{x}\right)^x $$

Since

$$ \lim_{x \to +\infty} \left(1 + \frac{1}{x}\right) = 1 $$

this expression has the indeterminate form $ 1^{ \infty } $

$$ \lim_{x \to +\infty} \left(1 + \frac{1}{x}\right)^x = 1^{ \infty } $$

To evaluate it, introduce an auxiliary variable $ y $:

$$ y = \left(1 + \frac{1}{x}\right)^x $$

Take the natural logarithm of both sides:

$$ \ln y = \ln\left(1 + \frac{1}{x}\right)^x $$

Using the properties of logarithms, we obtain

$$ \ln y = x \cdot \ln\left(1 + \frac{1}{x}\right) $$

Now introduce the change of variable $ t = \frac{1}{x} $, so that $ x = \frac{1}{t} $

$$ \ln y = \frac{1}{t} \cdot \ln(1+t) $$

$$ \ln y = \frac{\ln(1+t)}{t} $$

As $ x \to +\infty $, we have $ t \to 0^+ $. Hence, the limit becomes

$$ \lim_{x \to +\infty} \ln y = \lim_{t \to 0} \frac{\ln(1+t)}{t} $$

The latter is a standard limit: $ \lim_{t \to 0} \frac{\ln(1+t)}{t} = 1 $.

$$ \lim_{x \to +\infty} \ln y = 1 $$

To eliminate the natural logarithm, apply the exponential function to both sides:

$$ \lim_{x \to +\infty} e^{ \ln y } = e^1 $$

$$ \lim_{x \to +\infty} y = e^1 $$

Since the exponential function is continuous, it follows that

$$ \lim_{x \to +\infty} y = e^1 = e $$

Finally, substituting back $ y = \left(1 + \frac{1}{x}\right)^x $, we obtain

$$ \lim_{x \to +\infty} \left(1 + \frac{1}{x}\right)^x = e $$

as was to be proved.

And so on.