I came across the following limit in a math book
$\lim_{x\to \infty}\frac{(x^x)}{E(x)^{E(x)}} $
where $E(x)$ represent the floor function, and the question was to prove that this limit doesn't exist, in order to do that the following indication was given $ E(n+1/2)=n $ i am totally lost, and i don't have a clew how to proceed any help will be appreciated.
Hint: Write $f(x) = x^x/E(x)^{E(x)}$. Note that $E(x)$ is constant on intervals $[n,n+1)$ for integral $n$, and since $E(n) = n$ for integral $n$, we also have that $f(n)=1$. Thus $$f(1)=f(2)=f(3)=\cdots=f(n)=f(n+1)=\cdots$$ Now, can you show that $f(n +\frac12)$ is arbitarily large?