Interpreting limit notations My question is: Are the following notations equivalent or not:
$$(1)\;\;\;\;\;\;\text{When}\;||\textbf{x}||\rightarrow 0,\;\text{then}\;\;\;\frac{f(\textbf{x})}{||\textbf{x}||}\rightarrow0$$
$$(2)\;\;\;\;\;\;\lim_{||\textbf{x}||\rightarrow0} \frac{f(\textbf{x})}{||\textbf{x}||}=0\;\;?$$
The way I'm reading these are: 
$(1)$ When $||\textbf{x}||$ is approaching zero the fraction $\frac{f(\textbf{x})}{||\textbf{x}||}$ is also approaching zero. It will not never reach $0$, but it gets infinitely close to it. 
$(2)$ The limiting value of the fraction $\frac{f(\textbf{x})}{||\textbf{x}||}$ is zero when $||\textbf{x}||$ approaches zero. 
Are my interpretations correct or not? To me it seems that they are not exactly the same. The reason I ask this, because I got confused with a notation in one of my math books. I wasn't quite sure which one of $(1)$ or $(2)$ they meant. If they mean the same thing then my problem is solved :)  
In the above $f:\mathbb{R}^n\rightarrow \mathbb{R}$.

What if $f(x)=x^2$ and $x\in (0,\infty)$. Would the following two be equivalent: 
$$(1)\;\;\;|x|\rightarrow0,\;\;\; \frac{x^2}{x}=x\rightarrow0$$
$$(2)\;\;\;\;\;\;\;\lim_{|x|\rightarrow0} \frac{x^2}{x}=x=0$$
