Friday, September 23, 2016

calculus - Deconstructing 00





It is well known that 00 is an indeterminate form. One way to see that is noticing that


lim


yet,


\lim_{x\to0}\;x^0 = 1\quad.


What if we make both terms go to 0, that is, how much is


L = \lim_{x\to0^+}\;x^x\quad?


By taking x\in \langle 1/k\rangle_{k\in\mathbb{N*}}\,, I concluded that it equals \lim_{x\to\infty}\;x^{-1/x}, but that's not helpful.



Answer



This is, unfortunately, not very exciting. Rewrite x^x as e^{x\log x} and take that limit. One l'Hôpital later, you get 1.


No comments:

Post a Comment

analysis - Injection, making bijection

I have injection f \colon A \rightarrow B and I want to get bijection. Can I just resting codomain to f(A)? I know that every function i...