Floor function and little oh notation

Question

Can we replace $o([x]^a)$ where $[x]$ is floor of $x$ and $a$ is a positive number with $o(x^{a})$?

And can we replace $o(x^{a})$ with $o([x]^a)$?

Answer 1

$f \in o(g)$ is equivalent to $\lim_{x \to x_0} \left| \frac{f(x)}{g(x)} \right| = 0$, if this works with $g_1(x) = \lfloor x \rfloor^a$ does it also work with $g_2(x)=x^a$? It is also important to consider what $x_0$ you have to use.

Hint: Find an inequality between $g_1$ and $g_2$ and plug it into the limit for finding an lower/upper limit.