Understanding a step in a proof that $(a\times b)'=(a'\times b)+(a\times b')$.

Question

There is a step in this derivation of the derivative of the vector product of two differentiable functions that I do not understand: why is

\begin{split} \lim_{h\to 0} a(x+h)\times \frac{b(x+h)-b(x)}{h} = \left(\lim_{h\to 0}a(x+h)\right)\times \left(\lim_{h\to 0}\frac{b(x+h)-b(x)}{h}\right)? \end{split}

Answer 1

The cross product is a linear transformation, and it is therefore continuous as $a$ and $b$ are differentiable.

You can basically think of the cross product as a standard multiplication.