e

One definition of is that

Another definition of is that

There's also the definition of that it is the unique value satisfying

Let's show these are all equivalent.

The limit equals the series

Applying the binomial theorem to the first definition,

Expanding the binomial coefficient,

As , each factor in the product tends to , so each term tends to . Therefore

The power series for

Substituting in the limit definition gives , so

The same expansion applies where a lot of factors in the product approach as , leaving

Note that the substitution works even when is negative, because the limit definition of holds for as well. To see this, take the reciprocal of the original limit:

Taking the reciprocal again and substituting ,

Since the is absorbed into as , this gives .

The derivative of (from the series)

Differentiating the power series term by term,

The derivative of (from the limit)

We can also show this directly from the limit definition. Starting from the definition of a derivative,

So it suffices to show . Substituting and applying the first definition,

Therefore .

Solving

It is also possible to directly reason about what the differential equation says about how the function changes.

A rough first-order approximation of is .

Splitting this approximation into two steps gives applied twice, or .

Taking four steps instead gives applied four times, or .

Taking this to the limit gives , hence is a solution.