Chapter 12.1, Problem 84E

Proof Let r(t) and u(t) be vector-valued functions whose limits exist as $t\to c$. Prove that

$\underset{x\to c}{\lim }[r(t)\cdot u(t)]=\underset{x\to c}{\lim }r(t)\cdot \underset{x\to c}{\lim }u(t).$