Consider two vectors $x,y \in \mathbb{R}^n$ be parameterized by a value $t>0$, and suppose that
$\lim_{t \rightarrow 0} \frac{|x(t)|}{|y(t)|}=0,$
where $|\cdot|$ denotes the standard Euclidean norm. In other words as $t$ goes to zero the length of $x$ becomes insignificant relative to the norm of $y$.
I would like to claim then that
$\lim_{t \rightarrow 0} \frac{|x_i(t)|}{|y_i(t)|}=0,$
for any index $i \in 1, \ldots n$. In other words, the vanishing of the ratio of norms implies the vanishing of the ratio of corresponding elements.
BUT THIS STATEMENT IS FALSE -- SEE THE ANSWER BELOW!
Thanks for the help!