TITLE: Limit in n-dimensions with dot product
QUESTION [5 upvotes]: Let there be three vectors $\mathbf a$, $\mathbf b$, and $\mathbf h$ in $\mathbb R^n$. Define the real number
$$p(\mathbf h) \doteq (\mathbf a \cdot \mathbf h) (\mathbf b \cdot \mathbf h)$$
Does the following limit exist? In particular, does it equal $0$?
$$\lim_{\mathbf h \to \mathbf 0} \frac {p(\mathbf h)} {|\mathbf h|} $$
I was thinking of expressing $p$ as a product of sums, like
$$p(\mathbf h)=\left(\sum_{j=1}^n a_jh_j\right)\left(\sum_{k=1}^n b_kh_k\right) = \sum_{i=1}^na_ib_ih_i^2 + \sum_{j\neq k} a_jb_kh_jh_k $$
Are there any dot product properties I may be forgetting at this point?

REPLY [3 votes]: Observe that using Cauchy Schwarz Inequality yields:
$$0 \leq \bigg | \frac {p(h)} h \bigg | \leq \frac 1 {|h|} |a||b||h|^2 = |a||b||h| \to 0$$
as $h \to 0$. Hence by the Squeeze Thm your limit equals 0.