TITLE: Limit of $0^0$ (Indeterminate form)
QUESTION [8 upvotes]: Well this format of a limit $0^0$ is an indeterminate form.
I claim that whatever this limit is (which depends on the exact question) should always be in between $[0,1]$.
Is my claim correct?
I have no mathematical proof for it but just a basic idea, that any number base $0$ should try to pull towards $0$, whereas any number power $0$ should pull towards $1$.

REPLY [6 votes]: Your claim is wrong. You can choose any nonnegative number as the limit:
$$\lim_{x\to0^+}\left(e^{-1/x}\right)^{ax} = e^{-a}.$$
This example is from the Wikipedia article on $0^0$, as @PrasunBiswas suggested. There are other examples there showing a limit of infinity, etc.