There was a piece of information given in the “順帶二提" at the last post “Taylor’s polynomials@濟濟一堂", that is
then for any
This gave me the idea of setting the following standard short question Q.2 in the mock paper.
Show that is differentiable at .
May be we need to evaluate the following in the solution
By using l’ Hôpital’s rule directly, the situation ends with a ‘loop’ (just try).
Then I tried to prove by showing . To do so, I try to write a lower bound of , with just a bit expansion, I choose , then as . Afterwards, I think it is not a good method because we need the extra help ““, then I tried to do it by l’ Hôpital’s rule merely, just use the substitution , then the limit can be found easily, how amazing, isn’t it?