This is probably really obvious, but I am not seeing it.
Why does,
?
Apply L'Hopital's rule. By direct evaluation, it has the indeterminate form of
Let
We then see that . Now if we directly evaluate the limit, we have the case , which then satisfies the condition to apply L'Hopital's Rule.
Now,
But,
Therefore,
In a more general sense, I leave it for you to show that
Does this demystify things?
Just a little observation: the use of L'Hopital rule in demonstration that...
(1)
... presupposes that you have demonstrated before that...
(2)
... and that presupposes you have demonstrated before that...
(3)
... and that presupposes you have demonstrated before that...
(4)
... and that presupposes you have demonstrated before (1)...
So invoking L'Hopital's rule for demostration of (1) is a sort of 'vicious circle' and is better to avoid it...
Kind regards
you can't prove because it's just a definition! well, unless you have another definition of and you want to prove that the one you have implies the one that we have!
anyway, the only thing which needs to be proved here is that exists, which it does of course. now is just a name for the limit.
It is interesting however the demonstration that the number e can be defined as limit in two different ways...
From the pratical point of view the accurate computation of e as...
... requires an n of the order of several milions, while the accurate computation of e as...
... requires n less than 10! ...
Kind regards