The problem that is ocurring in your example is that the greatest integer function is not continuous. One consequence of this is that in general the limits $\lfloor \lim_{x\to c}f(x)\rfloor$ and $\lim_{x\to c}\lfloor f(x)\rfloor$ will not be equal to each other. This happens because the greatest integer function changes value abrubtly as you move along $\mathbb{R}$; every time you hit an integer, the value jumps up by $1$. So if, as $x\to c$, the function $f(x)$ approaches an integer $n$ from below, as in the case of $\frac{\sin{x}}{x}$ as $x\to 0$, the greatest integer function will (for $x$ close enough to $c$) return $n-1$, as $f(x)$ eventually exceeds $n-1$ as $x\to c$, but it never gets as high as $n$. On the other hand, if you take the limit first, you "attain" $n$, and then the greatest integer function returns $n$.
In general, $g(\lim_{x\to c}f(x))=\lim_{x\to c}(g(f(x))$ for all $c$ and $f$ such that $\lim_{x\to c}f(x)$ actually exists if and only if the function $g$ is continuous. This has several equivalent definitions, but can be thought of roughly as meaning the graph of $f$ has no sudden jumps (and doesn't oscillate too fast, like $\sin\frac{1}{x}$).
There are lots of big theorems about when you can swap the order of taking a limit and applying some other function, and when you can change the order of two limits etc. This turns up quite a lot with theorems about differentiation, integration and series, and mixtures of the three, as all are defined in terms of limits.
Hint: Assuming $b \ne 0$,
$$
\lim_{x \to 0} \frac{a}{x} \left\lfloor\frac{x}{b} \right\rfloor
= \lim_{x \to 0} \frac{a}{bx} \left\lfloor \frac{bx}{b}\right\rfloor
= \lim_{x \to 0} \frac{a}{b} \left( \frac{\lfloor x\rfloor}{x}\right)
= \frac{a}{b} \lim_{x \to 0} \frac{\lfloor x\rfloor}{x}
$$
For $x$ near $0$, $\lfloor x \rfloor$ is just $0$ on the right and $-1$ on the left.
Best Answer
Since $x-1\leq \lfloor x\rfloor\leq x$, for $x>0$ we have $$\frac{3x-3}{2x+1} \leq \frac{\lfloor 3x-2 \rfloor}{2x+1} \leq \frac{3x-2}{2x+1}$$ The limit of the left and right expressions as $x$ goes to infinity is $\frac 3 2$, so the same holds for the middle, by the squeeze theorem.