Let $Q_A,Q_B : \mathbb R^n \rightarrow \mathbb R$ be quadratic forms. Find a necessary and sufficient condition for $\lim_{\vec x \rightarrow \vec 0} \frac{Q_A(\vec x)}{Q_B(\vec x)}$ to exist in the strong sense.
I'm not really sure where to begin here. I wrote $\lim_{\vec x \rightarrow \vec 0} \frac{Q_A(\vec x)}{Q_B(\vec x)} = \lim_{\vec x \rightarrow \vec 0} \frac{\sum a_{ij}x_ix_j}{\sum b_{ij}x_ix_j}$, but I'm not sure how that helps.
Since we have $\vec x \rightarrow \vec 0$, we also have $x_i \rightarrow 0$. Can we find this condition component-wise?