Multivariable limits

Let's say we have a suspicion that $\lim\limits_{\vec{x} \to \vec{0}} f(\vec{x})=L=0$ for a function $f(\vec{x})$ such that after simplifying $\|f(\vec{x})-L\|<\varepsilon$, we get $\frac{1}{\|\vec{x}\|}<\varepsilon$.

What can we say about the existence of this limit?