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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