Greetings:
Is it necessarily true that the derivative of a limit is equal to the "appropriate" limit of the derivative? More specifically, I am trying to prove d/dx ex = ex as follows: It is readily shown that limit(n-->inf)[1+x/n]n = ex. Hence if my opening suggestion is true, then d/dx ex = limit[d/dx [1+x/n]n] = limit[1+x/n]n-1
= limit(n-->inf)[1+x/n]n / limit(n-->inf)[1+x/n]1 = ex/1 = ex.
i) Is the derivative of limit indeed equal to the limit of derivative?
ii) Are there suggestions as to how to prove i)? Please, suggest only; I'd like to put it all together on my own.
Thank you kindly.
Rich
Is it necessarily true that the derivative of a limit is equal to the "appropriate" limit of the derivative? More specifically, I am trying to prove d/dx ex = ex as follows: It is readily shown that limit(n-->inf)[1+x/n]n = ex. Hence if my opening suggestion is true, then d/dx ex = limit[d/dx [1+x/n]n] = limit[1+x/n]n-1
= limit(n-->inf)[1+x/n]n / limit(n-->inf)[1+x/n]1 = ex/1 = ex.
i) Is the derivative of limit indeed equal to the limit of derivative?
ii) Are there suggestions as to how to prove i)? Please, suggest only; I'd like to put it all together on my own.
Thank you kindly.
Rich