let
so if then
Thank you Moo, that is much simpler.
Here is another proof I thought of today. Taking the logarithmic derivative of , we get , and this goes to as . Hence the function satisfies for all complex , and it follows that
Edit : oops, this is exactly what Random Variable did
"To prove" means to argue for the validity of a statement based on prior knowledge. But what prior knowledge are we allowed to presuppose? In introductory courses, for example, this limit is sometimes used to define . Therefore, no prior knowledge of this function or its inverse may be used to show the existence of the limit in such a case.
Instead of involving logarithm and its development in the neighbourhood of 1, we can use (in a more "algebraic" way) the power series definition of the exponential: .
We develop .
For a fixed , the ratio tends to 1 (the numerator is a monic polynomial of degree in the variable ).
"Formally", we thus have .
The good point of this proof is that we can "see" why the limit is the exponential; this may have been Euler's way to prove the formula. However, it is not trivial to prove the limit rigorously (there is a summation in , not a fixed index...). This can probably be done elementarily, but here is a short not-so-elementary way. We have where for , and otherwise. For fixed , . The previous ratio is bounded by 1, hence for all . We have , therefore we can apply the bounded convergence theorem to take the limit in in the summation.
En passant: you probably meant little o's. Indeed, is a bounded sequence, and not a sequence converging toward (contrary to ). The first two lines were correct, but you couldn't deduce the limit with big O's.
If you use a function and try to find its derivative, by defining as the value of that makes we have...
.
For the function to be its own derivative, we would require that
and for simplicity, we will now use the symbol to represent this particular value of .
So we have
.
Now by letting , we notice that as .
So
Can you go from here?
But if you separate the limits then you may change the speeds of the variables, i.e., you may use instead of , which will lead you to a different result...
Also in (**), you can not take the power outside the limit either since it is not a constant, i.e., depends on the limit variable...
A simple note.
provided that .