OK, back to the issue of Euler’s number and the proof that it is . This is a proof based on, though expanded from, what I picked up from a rather excellent website here.

Once again we start from the proposition that there exists a number such that and hence (see here for why the second follows from the first).

Here we go…

At this point we still have , an indeterminate number, so we need to look for a determinate form.

So we take this as:

Now we have , another indeterminate, but we can also apply L’Hospital’s rule.

This states that, (a proof to follow sometime).

Here , and, of course, .

So, using the chain rule, , for the numerator, , : so

And for the denominator , giving for the whole thing,

Hence , so .

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