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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!

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1 Answer 1

It's not true. Consider, for $n = 2$, $x(t) = [t^2, t^2]$, $y(t) = [t^3,t]$. Then as $t \to 0$ $$\dfrac{\|x(t)\|}{\|y(t)\|} = \dfrac{\sqrt{2 t^4}}{\sqrt{t^6+t^2}} = \dfrac{\sqrt{2} t}{\sqrt{1+t^4}} \to 0 $$ but $$ \dfrac{x_1(t)}{y_1(t)} = \dfrac{t^2}{t^3} = \dfrac{1}{t} \ \text{diverges}$$

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Thanks Robert -- nice example. Guess I'll have to be more specific about the particular vectors $x$ and $y$ I'm considering (will post in a new question...). –  fuzzytron Aug 3 '12 at 19:37

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