A function $f$ is called regular varying with level $\alpha$ if for any $\lambda > 0$ it holds
$\lim_{x \rightarrow \infty} \frac{f(\lambda x)}{f(x)} = \lambda ^\alpha$.
A regular varying function with level $\alpha = 0$ is called slowly varying.
Given is following task:
Which of the following functions are regular varying?
$f_1(x) = e^{\log(x)}$
$f_2(x) = e^{\lfloor \log(x) \rfloor}$
Which of the following functions are slowly varying?
$f_3(x) = 2 + \sin(x)$
$f_4(x) = e^{(\log(x))^\beta}$, $\beta \in \mathbb R$
The first function is easy since I am interested in the limes for $x \rightarrow \infty$, I can use that for $x>0$ it holds that $f_1(x) = e^{\log(x)} = x$ and thus $\lim_{x \rightarrow \infty} \frac{f_1(\lambda x)}{f_1(x)} = \frac{\lambda x}{x} = \lambda$, so $f_1$ is regular varying with level $\alpha = 1$.
Do you have some hints for the other functions?
For $f_{2}(x)$, I would use the estimate $$ \log(x) - 1 \leq \lfloor \log(x) \rfloor \leq \log(x), $$ so $$ e^{-1} x \leq f_{2}(x) \leq x, $$ and $$ \log(\lambda) + \log(x) - 1 \leq \lfloor \log(x) \rfloor \leq \log(\lambda) + \log(x), $$ so $$ e^{-1} \lambda x \leq f_{2}(\lambda x) \leq \lambda x. $$
As for $f_3(x) = 2 + \sin(x)$, let $$\lambda = {1 \over 2}.$$ What is the behavior of $$ {f_{2}(\lambda x) \over f_{2}(x)}? $$ (Specifically, what are the minima and maxima?)